Theorem f1dmex 2819

Description: If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep 1075.

Assertion

Ref	Expression
f1dmex	⊢ (B ∈ C → (F:A–1-1→B → A ∈ V))

Proof of Theorem f1dmex

Step	Hyp	Ref	Expression
1		ssexg 1702	. . 3 ⊢ (B ∈ C → (ran F ⊆ B → ran F ∈ V))
2		f1f 2781	. . . 4 ⊢ (F:A–1-1→B → F:A–→B)
3		frn 2757	. . . 4 ⊢ (F:A–→B → ran F ⊆ B)
4	2, 3	syl 12	. . 3 ⊢ (F:A–1-1→B → ran F ⊆ B)
5	1, 4	syl5 22	. 2 ⊢ (B ∈ C → (F:A–1-1→B → ran F ∈ V))
6		fornex 2793	. . 3 ⊢ (ran F ∈ V → (◡F:ran F–onto→A → A ∈ V))
7		f1f1orn 2810	. . . 4 ⊢ (F:A–1-1→B → F:A–1-1-onto→ran F)
8		f1ocnv 2811	. . . 4 ⊢ (F:A–1-1-onto→ran F → ◡F:ran F–1-1-onto→A)
9		f1ofo 2806	. . . 4 ⊢ (◡F:ran F–1-1-onto→A → ◡F:ran F–onto→A)
10	7, 8, 9	3syl 21	. . 3 ⊢ (F:A–1-1→B → ◡F:ran F–onto→A)
11	6, 10	syl5 22	. 2 ⊢ (ran F ∈ V → (F:A–1-1→B → A ∈ V))
12	5, 11	syli 52	1 ⊢ (B ∈ C → (F:A–1-1→B → A ∈ V))

Syntax hints:   → wi 2   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ^◡ccnv 2409  ran crn 2411  –→wf 2418  –1-1→wf1 2419  –onto→wfo 2420  –1-1-onto→wf1o 2421

This theorem is referenced by:  abianfp 3000  f1dom2g 3300

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-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  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-id 2125  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

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