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Mirrors > Home > MPE Home > Th. List > cbvfo | Structured version Visualization version 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 6591 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
2 | cbvfo.1 | . . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | bicomd 225 | . . . . 5 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝜓 ↔ 𝜑)) |
4 | 3 | eqcoms 2829 | . . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → (𝜓 ↔ 𝜑)) |
5 | 4 | ralrn 6853 | . . 3 ⊢ (𝐹 Fn 𝐴 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
7 | forn 6592 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
8 | 7 | raleqdv 3415 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
9 | 6, 8 | bitr3d 283 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∀wral 3138 ran crn 5555 Fn wfn 6349 –onto→wfo 6352 ‘cfv 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fo 6360 df-fv 6362 |
This theorem is referenced by: cbvexfo 7045 cocan2 7047 f1oweALT 7672 supisolem 8936 qtopeu 22323 deg1leb 24688 dchrelbas4 25818 cnpconn 32477 cocanfo 34992 |
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