Theorem recseq 6399

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 5582	. . . . . . . 8 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑎 ↾ 𝑐)) = (𝐺‘(𝑎 ↾ 𝑐)))
2	1	eqeq2d 2218	. . . . . . 7 ⊢ (𝐹 = 𝐺 → ((𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)) ↔ (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))
3	2	ralbidv 2507	. . . . . 6 ⊢ (𝐹 = 𝐺 → (∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)) ↔ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))
4	3	anbi2d 464	. . . . 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 2508	. . . 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 2324	. . 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 3863	. 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 6398	. 2 ⊢ recs(𝐹) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
9		df-recs 6398	. 2 ⊢ recs(𝐺) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))}
10	7, 8, 9	3eqtr4g 2264	1 ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  {cab 2192  ∀wral 2485  ∃wrex 2486  ∪ cuni 3852  Oncon0 4414   ↾ cres 4681   Fn wfn 5271  ‘cfv 5276  recscrecs 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-uni 3853  df-br 4048  df-iota 5237  df-fv 5284  df-recs 6398

This theorem is referenced by:  rdgeq1  6464  rdgeq2  6465  freceq1  6485  freceq2  6486  frecsuclem  6499