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| Mirrors > Home > MPE Home > Th. List > mptcnv | Structured version Visualization version GIF version | ||
| Description: The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.) |
| Ref | Expression |
|---|---|
| mptcnv.1 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷))) |
| Ref | Expression |
|---|---|
| mptcnv | ⊢ (𝜑 → ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mptcnv.1 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷))) | |
| 2 | 1 | opabbidv 5209 | . 2 ⊢ (𝜑 → {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)}) |
| 3 | df-mpt 5226 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
| 4 | 3 | cnveqi 5885 | . . 3 ⊢ ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
| 5 | cnvopab 6157 | . . 3 ⊢ ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
| 6 | 4, 5 | eqtri 2765 | . 2 ⊢ ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
| 7 | df-mpt 5226 | . 2 ⊢ (𝑦 ∈ 𝐶 ↦ 𝐷) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)} | |
| 8 | 2, 6, 7 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {copab 5205 ↦ cmpt 5225 ◡ccnv 5684 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: nvocnv 7301 mptcnfimad 8011 mptfzshft 15814 pt1hmeo 23814 ballotlemrinv 34536 dssmapnvod 44033 |
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