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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 6534 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶)) | |
2 | 1 | 3adant3 1081 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶)) |
3 | fveq2 6191 | . . . . 5 ⊢ (𝐶 = (◡𝐹‘𝐷) → (𝐹‘𝐶) = (𝐹‘(◡𝐹‘𝐷))) | |
4 | 3 | eqcoms 2630 | . . . 4 ⊢ ((◡𝐹‘𝐷) = 𝐶 → (𝐹‘𝐶) = (𝐹‘(◡𝐹‘𝐷))) |
5 | f1ocnvfv2 6533 | . . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝐷)) = 𝐷) | |
6 | 5 | eqeq2d 2632 | . . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = (𝐹‘(◡𝐹‘𝐷)) ↔ (𝐹‘𝐶) = 𝐷)) |
7 | 4, 6 | syl5ib 234 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → ((◡𝐹‘𝐷) = 𝐶 → (𝐹‘𝐶) = 𝐷)) |
8 | 7 | 3adant2 1080 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((◡𝐹‘𝐷) = 𝐶 → (𝐹‘𝐶) = 𝐷)) |
9 | 2, 8 | impbid 202 | 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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ◡ccnv 5113 –1-1-onto→wf1o 5887 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 |
This theorem is referenced by: f1ofveu 6645 f1ocnvfv3 6646 1arith2 15632 f1omvdcnv 17864 f1omvdconj 17866 txhmeo 21606 iccpnfcnv 22743 dvcnvlem 23739 logeftb 24330 sqff1o 24908 bracnlnval 28973 cdlemg17h 35956 |
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