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Mirrors > Home > MPE Home > Th. List > csbcnv | Structured version Visualization version GIF version |
Description: Move class substitution in and out of the converse of a function. Version of csbcnvgALT 5229 without a sethood antecedent but depending on more axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (Revised by NM, 23-Aug-2018.) |
Ref | Expression |
---|---|
csbcnv | ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcbr 4637 | . . . 4 ⊢ ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦) | |
2 | 1 | opabbii 4649 | . . 3 ⊢ {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {〈𝑦, 𝑧〉 ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦} |
3 | csbopab 4932 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} | |
4 | df-cnv 5046 | . . 3 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = {〈𝑦, 𝑧〉 ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦} | |
5 | 2, 3, 4 | 3eqtr4ri 2643 | . 2 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} |
6 | df-cnv 5046 | . . 3 ⊢ ◡𝐹 = {〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} | |
7 | 6 | csbeq2i 3945 | . 2 ⊢ ⦋𝐴 / 𝑥⦌◡𝐹 = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} |
8 | 5, 7 | eqtr4i 2635 | 1 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 [wsbc 3402 ⦋csb 3499 class class class wbr 4583 {copab 4642 ◡ccnv 5037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-cnv 5046 |
This theorem is referenced by: esum2dlem 29481 csbpredg 32348 |
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