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Mirrors > Home > ILE Home > Th. List > cbvexfo | GIF version |
Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) |
Ref | Expression |
---|---|
cbvfo.1 | ⊢ ((𝐹‘𝑥) = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvexfo | ⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofn 5347 | . . 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 2142 | . . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → (𝜓 ↔ 𝜑)) |
5 | 4 | rexrn 5557 | . . 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 5348 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
8 | 7 | rexeqdv 2633 | . 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 1331 ∃wrex 2417 ran crn 4540 Fn wfn 5118 –onto→wfo 5121 ‘cfv 5123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fo 5129 df-fv 5131 |
This theorem is referenced by: (None) |
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