Theorem bnj1234 34552

Description: Technical lemma for bnj60 34601. 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 6633	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑑 ↔ 𝑔 Fn 𝑑))
2		fveq1 6883	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥))
3		reseq1 5968	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)))
4	3	opeq2d 4875	. . . . . . . . 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 2791	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → 𝑌 = 𝑍)
8	7	fveq2d 6888	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑌) = (𝐺‘𝑍))
9	2, 8	eqeq12d 2742	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑥) = (𝐺‘𝑌) ↔ (𝑔‘𝑥) = (𝐺‘𝑍)))
10	9	ralbidv 3171	. . . . 5 ⊢ (𝑓 = 𝑔 → (∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌) ↔ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)))
11	1, 10	anbi12d 630	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))))
12	11	rexbidv 3172	. . 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 2799	. 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 2764	1 ⊢ 𝐶 = 𝐷

Syntax hints:   ∧ wa 395   = wceq 1533  {cab 2703  ∀wral 3055  ∃wrex 3064  ⟨cop 4629   ↾ cres 5671   Fn wfn 6531  ‘cfv 6536   predc-bnj14 34227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is referenced by:  bnj1245  34553  bnj1256  34554  bnj1259  34555  bnj1296  34560  bnj1311  34563