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| Description: Alternate proof of fnex 7238, derived using the Axiom of Replacement in the form of funimaexg 6652. This version uses ax-pow 5364 and ax-un 7756, whereas fnex 7238 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| fnexALT | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fnrel 6669 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
| 2 | relssdmrn 6287 | . . . 4 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | |
| 3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | 
| 4 | 3 | adantr 480 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | 
| 5 | fndm 6670 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
| 6 | 5 | eleq1d 2825 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | 
| 7 | 6 | biimpar 477 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵) | 
| 8 | fnfun 6667 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
| 9 | funimaexg 6652 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V) | |
| 10 | 8, 9 | sylan 580 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V) | 
| 11 | imadmrn 6087 | . . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
| 12 | 5 | imaeq2d 6077 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴)) | 
| 13 | 11, 12 | eqtr3id 2790 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹 “ 𝐴)) | 
| 14 | 13 | eleq1d 2825 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (ran 𝐹 ∈ V ↔ (𝐹 “ 𝐴) ∈ V)) | 
| 15 | 14 | biimpar 477 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐹 “ 𝐴) ∈ V) → ran 𝐹 ∈ V) | 
| 16 | 10, 15 | syldan 591 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → ran 𝐹 ∈ V) | 
| 17 | xpexg 7771 | . . 3 ⊢ ((dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ V) → (dom 𝐹 × ran 𝐹) ∈ V) | |
| 18 | 7, 16, 17 | syl2anc 584 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (dom 𝐹 × ran 𝐹) ∈ V) | 
| 19 | ssexg 5322 | . 2 ⊢ ((𝐹 ⊆ (dom 𝐹 × ran 𝐹) ∧ (dom 𝐹 × ran 𝐹) ∈ V) → 𝐹 ∈ V) | |
| 20 | 4, 18, 19 | syl2anc 584 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 × cxp 5682 dom cdm 5684 ran crn 5685 “ cima 5687 Rel wrel 5689 Fun wfun 6554 Fn wfn 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-fun 6562 df-fn 6563 | 
| This theorem is referenced by: (None) | 
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