Theorem bnj1373 35427

Description: Technical lemma for bnj60 35459. 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 35419	. . . . . . 7 ⊢ (𝑓 ∈ 𝐵 → ∀𝑥 𝑓 ∈ 𝐵)
5	2, 4	bnj1307 35420	. . . . . 6 ⊢ (𝑓 ∈ 𝐶 → ∀𝑥 𝑓 ∈ 𝐶)
6	5	bnj1351 35223	. . . . 5 ⊢ ((𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∀𝑥(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
7	6	nf5i 2180	. . . 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 4598	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
10		bnj1318 35422	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
11	9, 10	uneq12d 4122	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
12	11	eqeq2d 2773	. . . . . 6 ⊢ (𝑥 = 𝑦 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
13	12	anbi2d 641	. . . . 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 286	. . . 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 3459	. 2 ⊢ ([𝑦 / 𝑥]𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
17	1, 16	bitri 278	1 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {cab 2740  ∀wral 3078  ∃wrex 3088  Vcvv 3454  [wsbc 3743   ∪ cun 3902   ⊆ wss 3904  {csn 4588  ⟨cop 4594  dom cdm 5660   ↾ cres 5662   Fn wfn 6531  ‘cfv 6536   predc-bnj14 35086   trClc-bnj18 35092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-iun 4957  df-br 5109  df-bnj14 35087  df-bnj18 35093

This theorem is used by:  bnj1374  35428  bnj1384  35429  bnj1398  35431  bnj1450  35447  bnj1489  35453