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Mirrors > Home > MPE Home > Th. List > f1ocnvfvb | Structured version Visualization version GIF version |
Description: Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.) |
Ref | Expression |
---|---|
f1ocnvfvb | ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = 𝐷 ↔ (◡𝐹‘𝐷) = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnvfv 7281 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶)) | |
2 | 1 | 3adant3 1129 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶)) |
3 | fveq2 6890 | . . . . 5 ⊢ (𝐶 = (◡𝐹‘𝐷) → (𝐹‘𝐶) = (𝐹‘(◡𝐹‘𝐷))) | |
4 | 3 | eqcoms 2733 | . . . 4 ⊢ ((◡𝐹‘𝐷) = 𝐶 → (𝐹‘𝐶) = (𝐹‘(◡𝐹‘𝐷))) |
5 | f1ocnvfv2 7280 | . . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝐷)) = 𝐷) | |
6 | 5 | eqeq2d 2736 | . . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = (𝐹‘(◡𝐹‘𝐷)) ↔ (𝐹‘𝐶) = 𝐷)) |
7 | 4, 6 | imbitrid 243 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → ((◡𝐹‘𝐷) = 𝐶 → (𝐹‘𝐶) = 𝐷)) |
8 | 7 | 3adant2 1128 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((◡𝐹‘𝐷) = 𝐶 → (𝐹‘𝐶) = 𝐷)) |
9 | 2, 8 | impbid 211 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = 𝐷 ↔ (◡𝐹‘𝐷) = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ◡ccnv 5669 –1-1-onto→wf1o 6540 ‘cfv 6541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5292 ax-nul 5299 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4317 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 |
This theorem is referenced by: f1ofveu 7408 f1ocnvfv3 7409 1arith2 16894 f1omvdcnv 19401 f1omvdconj 19403 rngqiprngu 21210 txhmeo 23723 iccpnfcnv 24885 dvcnvlem 25924 logeftb 26533 sqff1o 27130 bracnlnval 31940 cdlemg17h 40169 |
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