Theorem bnj1373 33731

Description: Technical lemma for bnj60 33763. 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
bnj1373.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1373.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1373.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1373.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1373.5	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)

Assertion

Ref	Expression
bnj1373	⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))

Distinct variable groups:   𝑥,𝐴   𝐵,𝑓   𝑥,𝑅   𝑓,𝑑,𝑥   𝑥,𝑦

Allowed substitution hints:   𝜏(𝑥,𝑦,𝑓,𝑑)   𝐴(𝑦,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑑)   𝑅(𝑦,𝑓,𝑑)   𝐺(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1373

Step	Hyp	Ref	Expression
1		bnj1373.5	. 2 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
2		bnj1373.3	. . . . . . 7 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
3		bnj1373.1	. . . . . . . 8 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
4	3	bnj1309 33723	. . . . . . 7 ⊢ (𝑓 ∈ 𝐵 → ∀𝑥 𝑓 ∈ 𝐵)
5	2, 4	bnj1307 33724	. . . . . 6 ⊢ (𝑓 ∈ 𝐶 → ∀𝑥 𝑓 ∈ 𝐶)
6	5	bnj1351 33527	. . . . 5 ⊢ ((𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∀𝑥(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
7	6	nf5i 2142	. . . 4 ⊢ Ⅎ𝑥(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
8		bnj1373.4	. . . . 5 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
9		sneq 4601	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
10		bnj1318 33726	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
11	9, 10	uneq12d 4129	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
12	11	eqeq2d 2742	. . . . . 6 ⊢ (𝑥 = 𝑦 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
13	12	anbi2d 629	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
14	8, 13	bitrid 282	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
15	7, 14	sbciegf 3781	. . 3 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
16	15	elv 3452	. 2 ⊢ ([𝑦 / 𝑥]𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
17	1, 16	bitri 274	1 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2708  ∀wral 3060  ∃wrex 3069  Vcvv 3446  [wsbc 3742   ∪ cun 3911   ⊆ wss 3913  {csn 4591  ⟨cop 4597  dom cdm 5638   ↾ cres 5640   Fn wfn 6496  ‘cfv 6501   predc-bnj14 33389   trClc-bnj18 33395

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-iun 4961  df-br 5111  df-bnj14 33390  df-bnj18 33396

This theorem is referenced by:  bnj1374  33732  bnj1384  33733  bnj1398  33735  bnj1450  33751  bnj1489  33757