Theorem predeq123 6290

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 5865	. . . . 5 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆)
3	2	3ad2ant1 1133	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → ◡𝑅 = ◡𝑆)
4		sneq 4632	. . . . 5 ⊢ (𝑋 = 𝑌 → {𝑋} = {𝑌})
5	4	3ad2ant3 1135	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → {𝑋} = {𝑌})
6	3, 5	imaeq12d 6050	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → (◡𝑅 “ {𝑋}) = (◡𝑆 “ {𝑌}))
7	1, 6	ineq12d 4209	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐵 ∩ (◡𝑆 “ {𝑌})))
8		df-pred 6289	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
9		df-pred 6289	. 2 ⊢ Pred(𝑆, 𝐵, 𝑌) = (𝐵 ∩ (◡𝑆 “ {𝑌}))
10	7, 8, 9	3eqtr4g 2796	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1541   ∩ cin 3943  {csn 4622  ^◡ccnv 5668   “ cima 5672  Predcpred 6288

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 2702

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 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289

This theorem is referenced by:  predeq1  6291  predeq2  6292  predeq3  6293  frecseq123  8249  wsuceq123  34616  wlimeq12  34621