Theorem hbiso 2930

Description: Bound-variable hypothesis builder for an isomorphism.

Hypotheses

Ref	Expression
hbiso.1	⊢ (y ∈ H → ∀x y ∈ H)
hbiso.2	⊢ (y ∈ R → ∀x y ∈ R)
hbiso.3	⊢ (y ∈ S → ∀x y ∈ S)
hbiso.4	⊢ (y ∈ A → ∀x y ∈ A)
hbiso.5	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

Ref	Expression
hbiso	⊢ (H Isom R, S (A, B) → ∀x H Isom R, S (A, B))

Distinct variable group(s):   y,H   y,R   y,S   y,A   y,B   x,y

Proof of Theorem hbiso

Step	Hyp	Ref	Expression
1		hbiso.1	. . . 4 ⊢ (y ∈ H → ∀x y ∈ H)
2		hbiso.4	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
3		hbiso.5	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
4	1, 2, 3	hbf1o 2798	. . 3 ⊢ (H:A–1-1-onto→B → ∀x H:A–1-1-onto→B)
5		ax-17 925	. . . . 5 ⊢ (y ∈ z → ∀x y ∈ z)
6	5, 2	hbel 1172	. . . 4 ⊢ (z ∈ A → ∀x z ∈ A)
7		ax-17 925	. . . . . 6 ⊢ (y ∈ w → ∀x y ∈ w)
8	7, 2	hbel 1172	. . . . 5 ⊢ (w ∈ A → ∀x w ∈ A)
9		hbiso.2	. . . . . . 7 ⊢ (y ∈ R → ∀x y ∈ R)
10	5, 9, 7	hbbr 2095	. . . . . 6 ⊢ (zRw → ∀x zRw)
11	1, 5	hbfv 2837	. . . . . . 7 ⊢ (y ∈ (H ‘z) → ∀x y ∈ (H ‘z))
12		hbiso.3	. . . . . . 7 ⊢ (y ∈ S → ∀x y ∈ S)
13	1, 7	hbfv 2837	. . . . . . 7 ⊢ (y ∈ (H ‘w) → ∀x y ∈ (H ‘w))
14	11, 12, 13	hbbr 2095	. . . . . 6 ⊢ ((H ‘z)S(H ‘w) → ∀x(H ‘z)S(H ‘w))
15	10, 14	hbbi 705	. . . . 5 ⊢ ((zRw ↔ (H ‘z)S(H ‘w)) → ∀x(zRw ↔ (H ‘z)S(H ‘w)))
16	8, 15	hbral 1236	. . . 4 ⊢ (∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w)) → ∀x∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w)))
17	6, 16	hbral 1236	. . 3 ⊢ (∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w)) → ∀x∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w)))
18	4, 17	hban 704	. 2 ⊢ ((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) → ∀x(H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))))
19		df-iso 2439	. 2 ⊢ (H Isom R, S (A, B) ↔ (H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))))
20	19	bial 695	. 2 ⊢ (∀x H Isom R, S (A, B) ↔ ∀x(H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))))
21	18, 19, 20	3imtr4 192	1 ⊢ (H Isom R, S (A, B) → ∀x H Isom R, S (A, B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   ∈ wel 803   ∈ wcel 1092  ∀wral 1201   class class class wbr 2054  –1-1-onto→wf1o 2421   ‘cfv 2422   Isom wiso 2423

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-iso 2439

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