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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 5214 | . 2 ⊢ (𝜑 → {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)}) |
3 | df-mpt 5232 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
4 | 3 | cnveqi 5888 | . . 3 ⊢ ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
5 | cnvopab 6160 | . . 3 ⊢ ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
6 | 4, 5 | eqtri 2763 | . 2 ⊢ ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
7 | df-mpt 5232 | . 2 ⊢ (𝑦 ∈ 𝐶 ↦ 𝐷) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)} | |
8 | 2, 6, 7 | 3eqtr4g 2800 | 1 ⊢ (𝜑 → ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {copab 5210 ↦ cmpt 5231 ◡ccnv 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5695 df-rel 5696 df-cnv 5697 |
This theorem is referenced by: nvocnv 7301 mptcnfimad 8010 mptfzshft 15811 pt1hmeo 23830 ballotlemrinv 34515 dssmapnvod 44010 |
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