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Mirrors > Home > MPE Home > Th. List > predeq123 | Structured version Visualization version GIF version |
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 13-Jun-2018.) |
Ref | Expression |
---|---|
predeq123 | ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1138 | . . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → 𝐴 = 𝐵) | |
2 | cnveq 5830 | . . . . 5 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆) | |
3 | 2 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → ◡𝑅 = ◡𝑆) |
4 | sneq 4597 | . . . . 5 ⊢ (𝑋 = 𝑌 → {𝑋} = {𝑌}) | |
5 | 4 | 3ad2ant3 1136 | . . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → {𝑋} = {𝑌}) |
6 | 3, 5 | imaeq12d 6015 | . . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → (◡𝑅 “ {𝑋}) = (◡𝑆 “ {𝑌})) |
7 | 1, 6 | ineq12d 4174 | . 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐵 ∩ (◡𝑆 “ {𝑌}))) |
8 | df-pred 6254 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
9 | df-pred 6254 | . 2 ⊢ Pred(𝑆, 𝐵, 𝑌) = (𝐵 ∩ (◡𝑆 “ {𝑌})) | |
10 | 7, 8, 9 | 3eqtr4g 2798 | 1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∩ cin 3910 {csn 4587 ◡ccnv 5633 “ cima 5637 Predcpred 6253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 |
This theorem is referenced by: predeq1 6256 predeq2 6257 predeq3 6258 frecseq123 8214 wsuceq123 34445 wlimeq12 34450 |
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