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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 5741 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 5097 | . . . . 5 ⊢ ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦) | |
2 | csbconstg 3821 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧) | |
3 | csbconstg 3821 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦) | |
4 | 2, 3 | breq12d 5056 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦)) |
5 | 1, 4 | syl5bb 286 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦)) |
6 | 5 | opabbidv 5109 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {〈𝑦, 𝑧〉 ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦}) |
7 | csbopabgALT 5426 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝑧𝐹𝑦}) | |
8 | df-cnv 5548 | . . . 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 2785 | . 2 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦}) |
11 | df-cnv 5548 | . . 3 ⊢ ◡𝐹 = {〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} | |
12 | 11 | csbeq2i 3810 | . 2 ⊢ ⦋𝐴 / 𝑥⦌◡𝐹 = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} |
13 | 10, 12 | eqtr4di 2792 | 1 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 [wsbc 3687 ⦋csb 3802 class class class wbr 5043 {copab 5105 ◡ccnv 5539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-cnv 5548 |
This theorem is referenced by: (None) |
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