Theorem wfrlem1 7301

Description: Lemma for well-founded recursion. The final item we are interested in is the union of acceptable functions 𝐵. This lemma just changes bound variables for later use. (Contributed by Scott Fenton, 21-Apr-2011.)

Hypothesis

Ref	Expression
wfrlem1.1	⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}

Assertion

Ref	Expression
wfrlem1	⊢ 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}

Distinct variable groups:   𝐴,𝑓,𝑔,𝑤,𝑥,𝑦,𝑧   𝑓,𝐹,𝑔,𝑤,𝑥,𝑦,𝑧   𝑅,𝑓,𝑔,𝑤,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔)

Proof of Theorem wfrlem1

Step	Hyp	Ref	Expression
1		wfrlem1.1	. 2 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2		fneq1 5893	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑥 ↔ 𝑔 Fn 𝑥))
3		fveq1 6102	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑦) = (𝑔‘𝑦))
4		reseq1 5311	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))
5	4	fveq2d 6107	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))
6	3, 5	eqeq12d 2625	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
7	6	ralbidv 2969	. . . . . 6 ⊢ (𝑓 = 𝑔 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
8	2, 7	3anbi13d 1393	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑔 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))))
9	8	exbidv 1837	. . . 4 ⊢ (𝑓 = 𝑔 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝑔 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))))
10		fneq2 5894	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑔 Fn 𝑥 ↔ 𝑔 Fn 𝑧))
11		sseq1 3589	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴))
12		sseq2 3590	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
13	12	raleqbi1dv 3123	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
14		predeq3 5601	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
15	14	sseq1d 3595	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧 ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))
16	15	cbvralv 3147	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧 ↔ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)
17	13, 16	syl6bb 275	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))
18	11, 17	anbi12d 743	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))
19		raleq 3115	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
20		fveq2 6103	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑔‘𝑦) = (𝑔‘𝑤))
21	14	reseq2d 5317	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))
22	21	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
23	20, 22	eqeq12d 2625	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
24	23	cbvralv 3147	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
25	19, 24	syl6bb 275	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
26	10, 18, 25	3anbi123d 1391	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑔 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))))
27	26	cbvexv 2263	. . . 4 ⊢ (∃𝑥(𝑔 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
28	9, 27	syl6bb 275	. . 3 ⊢ (𝑓 = 𝑔 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))))
29	28	cbvabv 2734	. 2 ⊢ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
30	1, 29	eqtri 2632	1 ⊢ 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}

Syntax hints:   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695  {cab 2596  ∀wral 2896   ⊆ wss 3540   ↾ cres 5040  Predcpred 5596   Fn wfn 5799  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812

This theorem is referenced by:  wfrlem2  7302  wfrlem3  7303  wfrlem3a  7304  wfrlem4  7305  wfrdmcl  7310