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Mirrors > Home > ILE Home > Th. List > cbvfo | GIF version |
Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
cbvfo.1 | ⊢ ((𝐹‘𝑥) = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvfo | ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofn 5305 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
2 | cbvfo.1 | . . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | bicomd 140 | . . . . 5 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝜓 ↔ 𝜑)) |
4 | 3 | eqcoms 2118 | . . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → (𝜓 ↔ 𝜑)) |
5 | 4 | ralrn 5512 | . . 3 ⊢ (𝐹 Fn 𝐴 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
6 | 1, 5 | syl 14 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
7 | forn 5306 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
8 | 7 | raleqdv 2606 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
9 | 6, 8 | bitr3d 189 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1314 ∀wral 2390 ran crn 4500 Fn wfn 5076 –onto→wfo 5079 ‘cfv 5081 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-sbc 2879 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-fo 5087 df-fv 5089 |
This theorem is referenced by: cocan2 5643 supisolem 6847 |
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