Theorem bnj149 32149

Description: Technical lemma for bnj151 32151. 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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj149.1	⊢ (𝜃1 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
bnj149.2	⊢ (𝜁0 ↔ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
bnj149.3	⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0)
bnj149.4	⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
bnj149.5	⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)
bnj149.6	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Assertion

Ref	Expression
bnj149	⊢ 𝜃1

Distinct variable groups:   𝐴,𝑓,𝑔,𝑥   𝑅,𝑓,𝑔,𝑥   𝑓,𝜁₁   𝑔,𝜁₀

Allowed substitution hints:   𝜑′(𝑥,𝑓,𝑔)   𝜓′(𝑥,𝑓,𝑔)   𝜁₀(𝑥,𝑓)   𝜑₁(𝑥,𝑓,𝑔)   𝜓₁(𝑥,𝑓,𝑔)   𝜃₁(𝑥,𝑓,𝑔)   𝜁₁(𝑥,𝑔)

Proof of Theorem bnj149

Step	Hyp	Ref	Expression
1		simpr1 1190	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓 Fn 1o)
2		df1o2 8118	. . . . . . . . 9 ⊢ 1o = {∅}
3	2	fneq2i 6453	. . . . . . . 8 ⊢ (𝑓 Fn 1o ↔ 𝑓 Fn {∅})
4	1, 3	sylib 220	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓 Fn {∅})
5		simpr2 1191	. . . . . . . . . 10 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝜑′)
6		bnj149.6	. . . . . . . . . 10 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
7	5, 6	sylib 220	. . . . . . . . 9 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
8		fvex 6685	. . . . . . . . . 10 ⊢ (𝑓‘∅) ∈ V
9	8	elsn 4584	. . . . . . . . 9 ⊢ ((𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
10	7, 9	sylibr 236	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
11		0ex 5213	. . . . . . . . 9 ⊢ ∅ ∈ V
12		fveq2 6672	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → (𝑓‘𝑔) = (𝑓‘∅))
13	12	eleq1d 2899	. . . . . . . . 9 ⊢ (𝑔 = ∅ → ((𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)}))
14	11, 13	ralsn 4621	. . . . . . . 8 ⊢ (∀𝑔 ∈ {∅} (𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
15	10, 14	sylibr 236	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → ∀𝑔 ∈ {∅} (𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)})
16		ffnfv 6884	. . . . . . 7 ⊢ (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓 Fn {∅} ∧ ∀𝑔 ∈ {∅} (𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)}))
17	4, 15, 16	sylanbrc 585	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)})
18		bnj93 32137	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
19	18	adantr 483	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → pred(𝑥, 𝐴, 𝑅) ∈ V)
20		fsng 6901	. . . . . . 7 ⊢ ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
21	11, 19, 20	sylancr 589	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
22	17, 21	mpbid 234	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
23	22	ex 415	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
24	23	alrimiv 1928	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∀𝑓((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
25		mo2icl 3707	. . 3 ⊢ (∀𝑓((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
26	24, 25	syl 17	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
27		bnj149.1	. 2 ⊢ (𝜃1 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
28	26, 27	mpbir 233	1 ⊢ 𝜃1

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1535   = wceq 1537   ∈ wcel 2114  ∃*wmo 2620  ∀wral 3140  Vcvv 3496  [wsbc 3774  ∅c0 4293  {csn 4569  ⟨cop 4575   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  1_oc1o 8097   predc-bnj14 31960   FrSe w-bnj15 31964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-1o 8104  df-bnj13 31963  df-bnj15 31965

This theorem is referenced by:  bnj151  32151