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Theorem bnj1373 32531
 Description: Technical lemma for bnj60 32563. 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 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1373.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1373.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1373.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1373.5 (𝜏′[𝑦 / 𝑥]𝜏)
Assertion
Ref Expression
bnj1373 (𝜏′ ↔ (𝑓𝐶 ∧ 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
StepHypRef Expression
1 bnj1373.5 . 2 (𝜏′[𝑦 / 𝑥]𝜏)
2 bnj1373.3 . . . . . . 7 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
3 bnj1373.1 . . . . . . . 8 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
43bnj1309 32523 . . . . . . 7 (𝑓𝐵 → ∀𝑥 𝑓𝐵)
52, 4bnj1307 32524 . . . . . 6 (𝑓𝐶 → ∀𝑥 𝑓𝐶)
65bnj1351 32327 . . . . 5 ((𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∀𝑥(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
76nf5i 2148 . . . 4 𝑥(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
8 bnj1373.4 . . . . 5 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
9 sneq 4533 . . . . . . . 8 (𝑥 = 𝑦 → {𝑥} = {𝑦})
10 bnj1318 32526 . . . . . . . 8 (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
119, 10uneq12d 4070 . . . . . . 7 (𝑥 = 𝑦 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
1211eqeq2d 2770 . . . . . 6 (𝑥 = 𝑦 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
1312anbi2d 632 . . . . 5 (𝑥 = 𝑦 → ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
148, 13syl5bb 286 . . . 4 (𝑥 = 𝑦 → (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
157, 14sbciegf 3735 . . 3 (𝑦 ∈ V → ([𝑦 / 𝑥]𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
1615elv 3416 . 2 ([𝑦 / 𝑥]𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
171, 16bitri 278 1 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  {cab 2736  ∀wral 3071  ∃wrex 3072  Vcvv 3410  [wsbc 3697   ∪ cun 3857   ⊆ wss 3859  {csn 4523  ⟨cop 4529  dom cdm 5525   ↾ cres 5527   Fn wfn 6331  ‘cfv 6336   predc-bnj14 32187   trClc-bnj18 32193 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-un 3864  df-sn 4524  df-pr 4526  df-op 4530  df-iun 4886  df-br 5034  df-bnj14 32188  df-bnj18 32194 This theorem is referenced by:  bnj1374  32532  bnj1384  32533  bnj1398  32535  bnj1450  32551  bnj1489  32557
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