f1dmex - Metamath Proof Explorer

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Theorem f1dmex 7939

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 5284. (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 6784	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	frnd 6722	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
3		ssexg 5322	. . . . 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 413	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐵 ∈ 𝐶 → ran 𝐹 ∈ V))
6		f1cnv 6854	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ^◡𝐹:ran 𝐹–1-1-onto→𝐴)
7		f1ofo 6837	. . . 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 7938	. . 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 407	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3947 ^◡ccnv 5674 ran crn 5676 –1-1→wf1 6537 –onto→wfo 6538 –1-1-onto→wf1o 6539

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548

This theorem is referenced by: f1ovv 7940 f1domg 8964 ordtypelem10 9518 oiexg 9526 inf3lem7 9625 pwfseqlem4 10653 pwfseqlem5 10654 grothomex 10820 gsumzf1o 19774 dprdf1o 19896 f1lindf 21368 tsmsf1o 23640 diophrw 41482

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