f1dmex - Metamath Proof Explorer

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Theorem f1dmex 7955

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 5249. (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 6774	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	frnd 6714	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
3		ssexg 5293	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → ran 𝐹 ∈ V)
4	2, 3	sylan 580	. . . 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 6842	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ^◡𝐹:ran 𝐹–1-1-onto→𝐴)
7		f1ofo 6825	. . . 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 7954	. . 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 2108 Vcvv 3459 ⊆ wss 3926 ^◡ccnv 5653 ran crn 5655 –1-1→wf1 6528 –onto→wfo 6529 –1-1-onto→wf1o 6530

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pr 5402 ax-un 7729

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539

This theorem is referenced by: f1ovv 7956 f1domg 8986 ordtypelem10 9541 oiexg 9549 inf3lem7 9648 pwfseqlem4 10676 pwfseqlem5 10677 grothomex 10843 gsumzf1o 19893 dprdf1o 20015 f1lindf 21782 tsmsf1o 24083 diophrw 42782

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