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| Mirrors > Home > MPE Home > Th. List > csbcnvgALT | Structured version Visualization version GIF version | ||
| Description: Move class substitution in and out of the converse of a relation. Version of csbcnv 5894 with a sethood antecedent but depending on fewer axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| csbcnvgALT | ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbcbr123 5197 | . . . . 5 ⊢ ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦) | |
| 2 | csbconstg 3918 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧) | |
| 3 | csbconstg 3918 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦) | |
| 4 | 2, 3 | breq12d 5156 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦)) |
| 5 | 1, 4 | bitrid 283 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦)) |
| 6 | 5 | opabbidv 5209 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {〈𝑦, 𝑧〉 ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦}) |
| 7 | csbopabgALT 5561 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝑧𝐹𝑦}) | |
| 8 | df-cnv 5693 | . . . 4 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = {〈𝑦, 𝑧〉 ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦} | |
| 9 | 8 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = {〈𝑦, 𝑧〉 ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦}) |
| 10 | 6, 7, 9 | 3eqtr4rd 2788 | . 2 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦}) |
| 11 | df-cnv 5693 | . . 3 ⊢ ◡𝐹 = {〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} | |
| 12 | 11 | csbeq2i 3907 | . 2 ⊢ ⦋𝐴 / 𝑥⦌◡𝐹 = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} |
| 13 | 10, 12 | eqtr4di 2795 | 1 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 [wsbc 3788 ⦋csb 3899 class class class wbr 5143 {copab 5205 ◡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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| 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-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-cnv 5693 |
| This theorem is referenced by: (None) |
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