Theorem fun11uni 2707

Description: The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function.

Assertion

Ref	Expression
fun11uni	⊢ (∀f ∈ A ((Fun f ∧ Fun ◡f) ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)) → (Fun ∪A ∧ Fun ◡∪A))

Distinct variable group(s):   f,g,A

Proof of Theorem fun11uni

Step	Hyp	Ref	Expression
1		pm3.26 256	. . . . 5 ⊢ ((Fun f ∧ Fun ◡f) → Fun f)
2	1	anim1i 269	. . . 4 ⊢ (((Fun f ∧ Fun ◡f) ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)) → (Fun f ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)))
3	2	r19.20si 1254	. . 3 ⊢ (∀f ∈ A ((Fun f ∧ Fun ◡f) ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)) → ∀f ∈ A (Fun f ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)))
4		fununi 2705	. . 3 ⊢ (∀f ∈ A (Fun f ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)) → Fun ∪A)
5	3, 4	syl 12	. 2 ⊢ (∀f ∈ A ((Fun f ∧ Fun ◡f) ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)) → Fun ∪A)
6		pm3.27 260	. . . . 5 ⊢ ((Fun f ∧ Fun ◡f) → Fun ◡f)
7	6	anim1i 269	. . . 4 ⊢ (((Fun f ∧ Fun ◡f) ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)) → (Fun ◡f ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)))
8	7	r19.20si 1254	. . 3 ⊢ (∀f ∈ A ((Fun f ∧ Fun ◡f) ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)) → ∀f ∈ A (Fun ◡f ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)))
9		funcnvuni 2706	. . 3 ⊢ (∀f ∈ A (Fun ◡f ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)) → Fun ◡∪A)
10	8, 9	syl 12	. 2 ⊢ (∀f ∈ A ((Fun f ∧ Fun ◡f) ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)) → Fun ◡∪A)
11	5, 10	jca 236	1 ⊢ (∀f ∈ A ((Fun f ∧ Fun ◡f) ∧ ∀g ∈ A (f ⊆ g ∨ g ⊆ f)) → (Fun ∪A ∧ Fun ◡∪A))

Syntax hints:   → wi 2   ∨ wo 195   ∧ wa 196  ∀wral 1201   ⊆ wss 1487  ∪cuni 1919  ^◡ccnv 2409  Fun wfun 2416

This theorem is referenced by:  infxpidmlem7 4939

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-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  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-iun 1996  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432

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