Theorem bnj1234 35195

Description: Technical lemma for bnj60 35244. 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 6593	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑑 ↔ 𝑔 Fn 𝑑))
2		fveq1 6843	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥))
3		reseq1 5942	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)))
4	3	opeq2d 4838	. . . . . . . . 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 2797	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → 𝑌 = 𝑍)
8	7	fveq2d 6848	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑌) = (𝐺‘𝑍))
9	2, 8	eqeq12d 2753	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑥) = (𝐺‘𝑌) ↔ (𝑔‘𝑥) = (𝐺‘𝑍)))
10	9	ralbidv 3161	. . . . 5 ⊢ (𝑓 = 𝑔 → (∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌) ↔ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)))
11	1, 10	anbi12d 633	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))))
12	11	rexbidv 3162	. . 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 2807	. 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 2770	1 ⊢ 𝐶 = 𝐷

Syntax hints:   ∧ wa 395   = wceq 1542  {cab 2715  ∀wral 3052  ∃wrex 3062  ⟨cop 4588   ↾ cres 5636   Fn wfn 6497  ‘cfv 6502   predc-bnj14 34871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6458  df-fun 6504  df-fn 6505  df-fv 6510

This theorem is referenced by:  bnj1245  35196  bnj1256  35197  bnj1259  35198  bnj1296  35203  bnj1311  35206