Theorem recseq 6203

Description: Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Assertion

Ref	Expression
recseq	⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))

Proof of Theorem recseq

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 5420	. . . . . . . 8 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑎 ↾ 𝑐)) = (𝐺‘(𝑎 ↾ 𝑐)))
2	1	eqeq2d 2151	. . . . . . 7 ⊢ (𝐹 = 𝐺 → ((𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)) ↔ (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))
3	2	ralbidv 2437	. . . . . 6 ⊢ (𝐹 = 𝐺 → (∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)) ↔ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))
4	3	anbi2d 459	. . . . 5 ⊢ (𝐹 = 𝐺 → ((𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))) ↔ (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))))
5	4	rexbidv 2438	. . . 4 ⊢ (𝐹 = 𝐺 → (∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))) ↔ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))))
6	5	abbidv 2257	. . 3 ⊢ (𝐹 = 𝐺 → {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} = {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))})
7	6	unieqd 3747	. 2 ⊢ (𝐹 = 𝐺 → ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))})
8		df-recs 6202	. 2 ⊢ recs(𝐹) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
9		df-recs 6202	. 2 ⊢ recs(𝐺) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))}
10	7, 8, 9	3eqtr4g 2197	1 ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331  {cab 2125  ∀wral 2416  ∃wrex 2417  ∪ cuni 3736  Oncon0 4285   ↾ cres 4541   Fn wfn 5118  ‘cfv 5123  recscrecs 6201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-recs 6202

This theorem is referenced by:  rdgeq1  6268  rdgeq2  6269  freceq1  6289  freceq2  6290  frecsuclem  6303