Theorem bnj1234 35210

Description: Technical lemma for bnj60 35259. 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
bnj1234.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1234.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1234.4	⊢ 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1234.5	⊢ 𝐷 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))}

Assertion

Ref	Expression
bnj1234	⊢ 𝐶 = 𝐷

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

Allowed substitution hints:   𝐴(𝑥,𝑓,𝑔,𝑑)   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑔,𝑑)   𝐷(𝑥,𝑓,𝑔,𝑑)   𝑅(𝑥,𝑓,𝑔,𝑑)   𝐺(𝑥,𝑑)   𝑌(𝑥,𝑓,𝑑)   𝑍(𝑥,𝑔,𝑑)

Proof of Theorem bnj1234

Step	Hyp	Ref	Expression
1		fneq1 6580	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑑 ↔ 𝑔 Fn 𝑑))
2		fveq1 6830	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥))
3		reseq1 5932	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)))
4	3	opeq2d 4814	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
5		bnj1234.2	. . . . . . . . 9 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
6		bnj1234.4	. . . . . . . . 9 ⊢ 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
7	4, 5, 6	3eqtr4g 2801	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → 𝑌 = 𝑍)
8	7	fveq2d 6835	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑌) = (𝐺‘𝑍))
9	2, 8	eqeq12d 2757	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑥) = (𝐺‘𝑌) ↔ (𝑔‘𝑥) = (𝐺‘𝑍)))
10	9	ralbidv 3164	. . . . 5 ⊢ (𝑓 = 𝑔 → (∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌) ↔ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)))
11	1, 10	anbi12d 639	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))))
12	11	rexbidv 3165	. . 3 ⊢ (𝑓 = 𝑔 → (∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))))
13	12	cbvabv 2811	. 2 ⊢ {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))}
14		bnj1234.3	. 2 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
15		bnj1234.5	. 2 ⊢ 𝐷 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))}
16	13, 14, 15	3eqtr4i 2774	1 ⊢ 𝐶 = 𝐷

Syntax hints:   ∧ wa 397   = wceq 1548  {cab 2719  ∀wral 3055  ∃wrex 3065  ⟨cop 4564   ↾ cres 5623   Fn wfn 6484  ‘cfv 6489   predc-bnj14 34886

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-res 5633  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497

This theorem is referenced by:  bnj1245  35211  bnj1256  35212  bnj1259  35213  bnj1296  35218  bnj1311  35221