Theorem predeq123 6298

Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 13-Jun-2018.)

Assertion

Ref	Expression
predeq123	⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))

Proof of Theorem predeq123

Step	Hyp	Ref	Expression
1		simp2 1137	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → 𝐴 = 𝐵)
2		cnveq 5871	. . . . 5 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆)
3	2	3ad2ant1 1133	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → ◡𝑅 = ◡𝑆)
4		sneq 4637	. . . . 5 ⊢ (𝑋 = 𝑌 → {𝑋} = {𝑌})
5	4	3ad2ant3 1135	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → {𝑋} = {𝑌})
6	3, 5	imaeq12d 6058	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → (◡𝑅 “ {𝑋}) = (◡𝑆 “ {𝑌}))
7	1, 6	ineq12d 4212	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐵 ∩ (◡𝑆 “ {𝑌})))
8		df-pred 6297	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
9		df-pred 6297	. 2 ⊢ Pred(𝑆, 𝐵, 𝑌) = (𝐵 ∩ (◡𝑆 “ {𝑌}))
10	7, 8, 9	3eqtr4g 2797	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1541   ∩ cin 3946  {csn 4627  ^◡ccnv 5674   “ cima 5678  Predcpred 6296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297

This theorem is referenced by:  predeq1  6299  predeq2  6300  predeq3  6301  frecseq123  8263  wsuceq123  34774  wlimeq12  34779