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Mirrors > Home > ILE Home > Th. List > f1ocnvfvb | 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 5596 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶)) | |
2 | 1 | 3adant3 966 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶)) |
3 | fveq2 5340 | . . . . 5 ⊢ (𝐶 = (◡𝐹‘𝐷) → (𝐹‘𝐶) = (𝐹‘(◡𝐹‘𝐷))) | |
4 | 3 | eqcoms 2098 | . . . 4 ⊢ ((◡𝐹‘𝐷) = 𝐶 → (𝐹‘𝐶) = (𝐹‘(◡𝐹‘𝐷))) |
5 | f1ocnvfv2 5595 | . . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝐷)) = 𝐷) | |
6 | 5 | eqeq2d 2106 | . . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = (𝐹‘(◡𝐹‘𝐷)) ↔ (𝐹‘𝐶) = 𝐷)) |
7 | 4, 6 | syl5ib 153 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → ((◡𝐹‘𝐷) = 𝐶 → (𝐹‘𝐶) = 𝐷)) |
8 | 7 | 3adant2 965 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((◡𝐹‘𝐷) = 𝐶 → (𝐹‘𝐶) = 𝐷)) |
9 | 2, 8 | impbid 128 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = 𝐷 ↔ (◡𝐹‘𝐷) = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 927 = wceq 1296 ∈ wcel 1445 ◡ccnv 4466 –1-1-onto→wf1o 5048 ‘cfv 5049 |
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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 |
This theorem is referenced by: f1ofveu 5678 f1ocnvfv3 5679 seq3f1olemstep 10067 |
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