f1dmex - Metamath Proof Explorer

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Theorem f1dmex 7901

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 5223. (Contributed by NM, 4-Sep-2004.)

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

**Proof of Theorem f1dmex**
Step	Hyp	Ref	Expression
1		f1f 6729	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	frnd 6669	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
3		ssexg 5267	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → ran 𝐹 ∈ V)
4	2, 3	sylan 581	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → ran 𝐹 ∈ V)
5	4	ex 412	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐵 ∈ 𝐶 → ran 𝐹 ∈ V))
6		f1cnv 6797	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ^◡𝐹:ran 𝐹–1-1-onto→𝐴)
7		f1ofo 6780	. . . 4 ⊢ (^◡𝐹:ran 𝐹–1-1-onto→𝐴 → ^◡𝐹:ran 𝐹–onto→𝐴)
8	6, 7	syl 17	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → ^◡𝐹:ran 𝐹–onto→𝐴)
9		focdmex 7900	. . 3 ⊢ (ran 𝐹 ∈ V → (^◡𝐹:ran 𝐹–onto→𝐴 → 𝐴 ∈ V))
10	5, 8, 9	syl6ci 71	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐵 ∈ 𝐶 → 𝐴 ∈ V))
11	10	imp 406	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2114 Vcvv 3439 ⊆ wss 3900 ^◡ccnv 5622 ran crn 5624 –1-1→wf1 6488 –onto→wfo 6489 –1-1-onto→wf1o 6490

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pr 5376 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4285 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4947 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499

This theorem is referenced by: f1ovv 7902 f1domg 8910 ordtypelem10 9434 oiexg 9442 inf3lem7 9545 pwfseqlem4 10575 pwfseqlem5 10576 grothomex 10742 gsumzf1o 19843 dprdf1o 19965 f1lindf 21779 tsmsf1o 24091 onvf1od 35280 diophrw 43038

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