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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 relation. Version of csbcnvgALT 5898 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 5203 | . . . 4 ⊢ ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦) | |
2 | 1 | opabbii 5215 | . . 3 ⊢ {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {〈𝑦, 𝑧〉 ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦} |
3 | csbopab 5565 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} | |
4 | df-cnv 5697 | . . 3 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = {〈𝑦, 𝑧〉 ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦} | |
5 | 2, 3, 4 | 3eqtr4ri 2774 | . 2 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} |
6 | df-cnv 5697 | . . 3 ⊢ ◡𝐹 = {〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} | |
7 | 6 | csbeq2i 3916 | . 2 ⊢ ⦋𝐴 / 𝑥⦌◡𝐹 = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} |
8 | 5, 7 | eqtr4i 2766 | 1 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 [wsbc 3791 ⦋csb 3908 class class class wbr 5148 {copab 5210 ◡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-10 2139 ax-11 2155 ax-12 2175 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-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 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-cnv 5697 |
This theorem is referenced by: csbpredg 6329 esum2dlem 34073 |
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