Theorem bnj1296 32714

Description: Technical lemma for bnj60 32755. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1296.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1296.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1296.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1296.4	⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)
bnj1296.5	⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
bnj1296.6	⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
bnj1296.7	⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))
bnj1296.18	⊢ (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = (ℎ ↾ pred(𝑥, 𝐴, 𝑅)))
bnj1296.9	⊢ 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1296.10	⊢ 𝐾 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))}
bnj1296.11	⊢ 𝑊 = ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1296.12	⊢ 𝐿 = {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑊))}

Assertion

Ref	Expression
bnj1296	⊢ (𝜓 → (𝑔‘𝑥) = (ℎ‘𝑥))

Distinct variable groups:   𝐵,𝑓,𝑔   𝐵,ℎ,𝑓   𝑥,𝐷   𝐺,𝑑,𝑓,𝑔   ℎ,𝐺,𝑑   𝑊,𝑑,𝑓   𝑔,𝑌   ℎ,𝑌   𝑍,𝑑,𝑓   𝑥,𝑑,𝑓,𝑔   𝑥,ℎ

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝜓(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐴(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐷(𝑦,𝑓,𝑔,ℎ,𝑑)   𝑅(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐸(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐺(𝑥,𝑦)   𝐾(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐿(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝑊(𝑥,𝑦,𝑔,ℎ)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑔,ℎ)

Proof of Theorem bnj1296

Step	Hyp	Ref	Expression
1		bnj1296.18	. . . . 5 ⊢ (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = (ℎ ↾ pred(𝑥, 𝐴, 𝑅)))
2	1	opeq2d 4791	. . . 4 ⊢ (𝜓 → ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩)
3		bnj1296.9	. . . 4 ⊢ 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
4		bnj1296.11	. . . 4 ⊢ 𝑊 = ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩
5	2, 3, 4	3eqtr4g 2803	. . 3 ⊢ (𝜓 → 𝑍 = 𝑊)
6	5	fveq2d 6721	. 2 ⊢ (𝜓 → (𝐺‘𝑍) = (𝐺‘𝑊))
7		bnj1296.7	. . . 4 ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))
8		bnj1296.6	. . . . 5 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
9		bnj1296.10	. . . . . . . . . . 11 ⊢ 𝐾 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))}
10	9	bnj1436 32532	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐾 → ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)))
11		fndm 6481	. . . . . . . . . . 11 ⊢ (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
12	11	anim1i 618	. . . . . . . . . 10 ⊢ ((𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)) → (dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)))
13	10, 12	bnj31 32410	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝐾 → ∃𝑑 ∈ 𝐵 (dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)))
14		raleq 3319	. . . . . . . . . . 11 ⊢ (dom 𝑔 = 𝑑 → (∀𝑥 ∈ dom 𝑔(𝑔‘𝑥) = (𝐺‘𝑍) ↔ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)))
15	14	pm5.32i 578	. . . . . . . . . 10 ⊢ ((dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ dom 𝑔(𝑔‘𝑥) = (𝐺‘𝑍)) ↔ (dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)))
16	15	rexbii 3170	. . . . . . . . 9 ⊢ (∃𝑑 ∈ 𝐵 (dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ dom 𝑔(𝑔‘𝑥) = (𝐺‘𝑍)) ↔ ∃𝑑 ∈ 𝐵 (dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)))
17	13, 16	sylibr 237	. . . . . . . 8 ⊢ (𝑔 ∈ 𝐾 → ∃𝑑 ∈ 𝐵 (dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ dom 𝑔(𝑔‘𝑥) = (𝐺‘𝑍)))
18		simpr 488	. . . . . . . 8 ⊢ ((dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ dom 𝑔(𝑔‘𝑥) = (𝐺‘𝑍)) → ∀𝑥 ∈ dom 𝑔(𝑔‘𝑥) = (𝐺‘𝑍))
19	17, 18	bnj31 32410	. . . . . . 7 ⊢ (𝑔 ∈ 𝐾 → ∃𝑑 ∈ 𝐵 ∀𝑥 ∈ dom 𝑔(𝑔‘𝑥) = (𝐺‘𝑍))
20	19	bnj1265 32505	. . . . . 6 ⊢ (𝑔 ∈ 𝐾 → ∀𝑥 ∈ dom 𝑔(𝑔‘𝑥) = (𝐺‘𝑍))
21		bnj1296.2	. . . . . . 7 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
22		bnj1296.3	. . . . . . 7 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
23	21, 22, 3, 9	bnj1234 32706	. . . . . 6 ⊢ 𝐶 = 𝐾
24	20, 23	eleq2s 2856	. . . . 5 ⊢ (𝑔 ∈ 𝐶 → ∀𝑥 ∈ dom 𝑔(𝑔‘𝑥) = (𝐺‘𝑍))
25	8, 24	bnj770 32455	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ dom 𝑔(𝑔‘𝑥) = (𝐺‘𝑍))
26	7, 25	bnj835 32451	. . 3 ⊢ (𝜓 → ∀𝑥 ∈ dom 𝑔(𝑔‘𝑥) = (𝐺‘𝑍))
27		bnj1296.4	. . . . 5 ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)
28	27	bnj1292 32508	. . . 4 ⊢ 𝐷 ⊆ dom 𝑔
29		bnj1296.5	. . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
30	29, 7	bnj1212 32492	. . . 4 ⊢ (𝜓 → 𝑥 ∈ 𝐷)
31	28, 30	bnj1213 32491	. . 3 ⊢ (𝜓 → 𝑥 ∈ dom 𝑔)
32	26, 31	bnj1294 32510	. 2 ⊢ (𝜓 → (𝑔‘𝑥) = (𝐺‘𝑍))
33		bnj1296.12	. . . . . . . . . . 11 ⊢ 𝐿 = {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑊))}
34	33	bnj1436 32532	. . . . . . . . . 10 ⊢ (ℎ ∈ 𝐿 → ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑊)))
35		fndm 6481	. . . . . . . . . . 11 ⊢ (ℎ Fn 𝑑 → dom ℎ = 𝑑)
36	35	anim1i 618	. . . . . . . . . 10 ⊢ ((ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑊)) → (dom ℎ = 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑊)))
37	34, 36	bnj31 32410	. . . . . . . . 9 ⊢ (ℎ ∈ 𝐿 → ∃𝑑 ∈ 𝐵 (dom ℎ = 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑊)))
38		raleq 3319	. . . . . . . . . . 11 ⊢ (dom ℎ = 𝑑 → (∀𝑥 ∈ dom ℎ(ℎ‘𝑥) = (𝐺‘𝑊) ↔ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑊)))
39	38	pm5.32i 578	. . . . . . . . . 10 ⊢ ((dom ℎ = 𝑑 ∧ ∀𝑥 ∈ dom ℎ(ℎ‘𝑥) = (𝐺‘𝑊)) ↔ (dom ℎ = 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑊)))
40	39	rexbii 3170	. . . . . . . . 9 ⊢ (∃𝑑 ∈ 𝐵 (dom ℎ = 𝑑 ∧ ∀𝑥 ∈ dom ℎ(ℎ‘𝑥) = (𝐺‘𝑊)) ↔ ∃𝑑 ∈ 𝐵 (dom ℎ = 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑊)))
41	37, 40	sylibr 237	. . . . . . . 8 ⊢ (ℎ ∈ 𝐿 → ∃𝑑 ∈ 𝐵 (dom ℎ = 𝑑 ∧ ∀𝑥 ∈ dom ℎ(ℎ‘𝑥) = (𝐺‘𝑊)))
42		simpr 488	. . . . . . . 8 ⊢ ((dom ℎ = 𝑑 ∧ ∀𝑥 ∈ dom ℎ(ℎ‘𝑥) = (𝐺‘𝑊)) → ∀𝑥 ∈ dom ℎ(ℎ‘𝑥) = (𝐺‘𝑊))
43	41, 42	bnj31 32410	. . . . . . 7 ⊢ (ℎ ∈ 𝐿 → ∃𝑑 ∈ 𝐵 ∀𝑥 ∈ dom ℎ(ℎ‘𝑥) = (𝐺‘𝑊))
44	43	bnj1265 32505	. . . . . 6 ⊢ (ℎ ∈ 𝐿 → ∀𝑥 ∈ dom ℎ(ℎ‘𝑥) = (𝐺‘𝑊))
45	21, 22, 4, 33	bnj1234 32706	. . . . . 6 ⊢ 𝐶 = 𝐿
46	44, 45	eleq2s 2856	. . . . 5 ⊢ (ℎ ∈ 𝐶 → ∀𝑥 ∈ dom ℎ(ℎ‘𝑥) = (𝐺‘𝑊))
47	8, 46	bnj771 32456	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ dom ℎ(ℎ‘𝑥) = (𝐺‘𝑊))
48	7, 47	bnj835 32451	. . 3 ⊢ (𝜓 → ∀𝑥 ∈ dom ℎ(ℎ‘𝑥) = (𝐺‘𝑊))
49	27	bnj1293 32509	. . . 4 ⊢ 𝐷 ⊆ dom ℎ
50	49, 30	bnj1213 32491	. . 3 ⊢ (𝜓 → 𝑥 ∈ dom ℎ)
51	48, 50	bnj1294 32510	. 2 ⊢ (𝜓 → (ℎ‘𝑥) = (𝐺‘𝑊))
52	6, 32, 51	3eqtr4d 2787	1 ⊢ (𝜓 → (𝑔‘𝑥) = (ℎ‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3062  {crab 3065   ∩ cin 3865   ⊆ wss 3866  ⟨cop 4547   class class class wbr 5053  dom cdm 5551   ↾ cres 5553   Fn wfn 6375  ‘cfv 6380   ∧ w-bnj17 32377   predc-bnj14 32379   FrSe w-bnj15 32383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-res 5563  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388  df-bnj17 32378

This theorem is referenced by:  bnj1311  32717