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Theorem bnj1373 35044
Description: Technical lemma for bnj60 35076. 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 35036 . . . . . . 7 (𝑓𝐵 → ∀𝑥 𝑓𝐵)
52, 4bnj1307 35037 . . . . . 6 (𝑓𝐶 → ∀𝑥 𝑓𝐶)
65bnj1351 34840 . . . . 5 ((𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∀𝑥(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
76nf5i 2146 . . . 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 4636 . . . . . . . 8 (𝑥 = 𝑦 → {𝑥} = {𝑦})
10 bnj1318 35039 . . . . . . . 8 (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
119, 10uneq12d 4169 . . . . . . 7 (𝑥 = 𝑦 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
1211eqeq2d 2748 . . . . . 6 (𝑥 = 𝑦 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
1312anbi2d 630 . . . . 5 (𝑥 = 𝑦 → ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
148, 13bitrid 283 . . . 4 (𝑥 = 𝑦 → (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
157, 14sbciegf 3827 . . 3 (𝑦 ∈ V → ([𝑦 / 𝑥]𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
1615elv 3485 . 2 ([𝑦 / 𝑥]𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
171, 16bitri 275 1 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  [wsbc 3788  cun 3949  wss 3951  {csn 4626  cop 4632  dom cdm 5685  cres 5687   Fn wfn 6556  cfv 6561   predc-bnj14 34702   trClc-bnj18 34708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-iun 4993  df-br 5144  df-bnj14 34703  df-bnj18 34709
This theorem is referenced by:  bnj1374  35045  bnj1384  35046  bnj1398  35048  bnj1450  35064  bnj1489  35070
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