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Mirrors > Home > MPE Home > Th. List > f1ocnvfv3 | Structured version Visualization version GIF version |
Description: Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
f1ocnvfv3 | ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) = (℩𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnvdm 7033 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) ∈ 𝐴) | |
2 | f1ocnvfvb 7028 | . . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥)) | |
3 | 2 | 3expa 1113 | . . . . 5 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥)) |
4 | 3 | an32s 650 | . . . 4 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥)) |
5 | eqcom 2826 | . . . 4 ⊢ (𝑥 = (◡𝐹‘𝐶) ↔ (◡𝐹‘𝐶) = 𝑥) | |
6 | 4, 5 | syl6bbr 291 | . . 3 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ 𝑥 = (◡𝐹‘𝐶))) |
7 | 1, 6 | riota5 7135 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (℩𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶) = (◡𝐹‘𝐶)) |
8 | 7 | eqcomd 2825 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) = (℩𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ◡ccnv 5547 –1-1-onto→wf1o 6347 ‘cfv 6348 ℩crio 7105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 |
This theorem is referenced by: (None) |
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