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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 1133 | . . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → 𝐴 = 𝐵) | |
2 | cnveq 5738 | . . . . 5 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆) | |
3 | 2 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → ◡𝑅 = ◡𝑆) |
4 | sneq 4570 | . . . . 5 ⊢ (𝑋 = 𝑌 → {𝑋} = {𝑌}) | |
5 | 4 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → {𝑋} = {𝑌}) |
6 | 3, 5 | imaeq12d 5924 | . . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → (◡𝑅 “ {𝑋}) = (◡𝑆 “ {𝑌})) |
7 | 1, 6 | ineq12d 4189 | . 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐵 ∩ (◡𝑆 “ {𝑌}))) |
8 | df-pred 6142 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
9 | df-pred 6142 | . 2 ⊢ Pred(𝑆, 𝐵, 𝑌) = (𝐵 ∩ (◡𝑆 “ {𝑌})) | |
10 | 7, 8, 9 | 3eqtr4g 2881 | 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 1083 = wceq 1533 ∩ cin 3934 {csn 4560 ◡ccnv 5548 “ cima 5552 Predcpred 6141 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 |
This theorem is referenced by: predeq1 6144 predeq2 6145 predeq3 6146 wsuceq123 33096 wlimeq12 33101 frecseq123 33114 |
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