Theorem recseq 6364

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 5557	. . . . . . . 8 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑎 ↾ 𝑐)) = (𝐺‘(𝑎 ↾ 𝑐)))
2	1	eqeq2d 2208	. . . . . . 7 ⊢ (𝐹 = 𝐺 → ((𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)) ↔ (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))
3	2	ralbidv 2497	. . . . . 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 2498	. . . 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 2314	. . 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 3850	. 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 6363	. 2 ⊢ recs(𝐹) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
9		df-recs 6363	. 2 ⊢ recs(𝐺) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))}
10	7, 8, 9	3eqtr4g 2254	1 ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  {cab 2182  ∀wral 2475  ∃wrex 2476  ∪ cuni 3839  Oncon0 4398   ↾ cres 4665   Fn wfn 5253  ‘cfv 5258  recscrecs 6362

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-recs 6363

This theorem is referenced by:  rdgeq1  6429  rdgeq2  6430  freceq1  6450  freceq2  6451  frecsuclem  6464