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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 4716 | . 2 ⊢ (𝜑 → {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)}) |
3 | df-mpt 4730 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
4 | 3 | cnveqi 5297 | . . 3 ⊢ ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
5 | cnvopab 5533 | . . 3 ⊢ ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
6 | 4, 5 | eqtri 2644 | . 2 ⊢ ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
7 | df-mpt 4730 | . 2 ⊢ (𝑦 ∈ 𝐶 ↦ 𝐷) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)} | |
8 | 2, 6, 7 | 3eqtr4g 2681 | 1 ⊢ (𝜑 → ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {copab 4712 ↦ cmpt 4729 ◡ccnv 5113 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 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-rab 2921 df-v 3202 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-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-rel 5121 df-cnv 5122 |
This theorem is referenced by: nvocnv 6537 mptfzshft 14510 fsumrev 14511 fprodrev 14707 pt1hmeo 21609 ballotlemrinv 30595 dssmapnvod 38314 |
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