Theorem unxpdomlem2 8717

Description: Lemma for unxpdom 8719. (Contributed by Mario Carneiro, 13-Jan-2013.)

Hypotheses

Ref	Expression
unxpdomlem1.1	⊢ 𝐹 = (𝑥 ∈ (𝑎 ∪ 𝑏) ↦ 𝐺)
unxpdomlem1.2	⊢ 𝐺 = if(𝑥 ∈ 𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
unxpdomlem2.1	⊢ (𝜑 → 𝑤 ∈ (𝑎 ∪ 𝑏))
unxpdomlem2.2	⊢ (𝜑 → ¬ 𝑚 = 𝑛)
unxpdomlem2.3	⊢ (𝜑 → ¬ 𝑠 = 𝑡)

Assertion

Ref	Expression
unxpdomlem2	⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → ¬ (𝐹‘𝑧) = (𝐹‘𝑤))

Distinct variable groups:   𝑤,𝐹,𝑧   𝑎,𝑏,𝑚,𝑛,𝑠,𝑡,𝑤,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑧,𝑤,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)   𝐹(𝑥,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)   𝐺(𝑥,𝑧,𝑤,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)

Proof of Theorem unxpdomlem2

Step	Hyp	Ref	Expression
1		unxpdomlem2.3	. . 3 ⊢ (𝜑 → ¬ 𝑠 = 𝑡)
2	1	adantr 483	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → ¬ 𝑠 = 𝑡)
3		elun1 4152	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑎 → 𝑧 ∈ (𝑎 ∪ 𝑏))
4	3	ad2antrl 726	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → 𝑧 ∈ (𝑎 ∪ 𝑏))
5		unxpdomlem1.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ (𝑎 ∪ 𝑏) ↦ 𝐺)
6		unxpdomlem1.2	. . . . . . . . . 10 ⊢ 𝐺 = if(𝑥 ∈ 𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
7	5, 6	unxpdomlem1 8716	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑎 ∪ 𝑏) → (𝐹‘𝑧) = if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
8	4, 7	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → (𝐹‘𝑧) = if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
9		iftrue 4473	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑎 → if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
10	9	ad2antrl 726	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
11	8, 10	eqtrd 2856	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → (𝐹‘𝑧) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
12		unxpdomlem2.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑤 ∈ (𝑎 ∪ 𝑏))
13	12	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → 𝑤 ∈ (𝑎 ∪ 𝑏))
14	5, 6	unxpdomlem1 8716	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝑎 ∪ 𝑏) → (𝐹‘𝑤) = if(𝑤 ∈ 𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
15	13, 14	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → (𝐹‘𝑤) = if(𝑤 ∈ 𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
16		iffalse 4476	. . . . . . . . 9 ⊢ (¬ 𝑤 ∈ 𝑎 → if(𝑤 ∈ 𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
17	16	ad2antll 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → if(𝑤 ∈ 𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
18	15, 17	eqtrd 2856	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → (𝐹‘𝑤) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
19	11, 18	eqeq12d 2837	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
20	19	biimpa 479	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
21		vex 3498	. . . . . 6 ⊢ 𝑧 ∈ V
22		vex 3498	. . . . . . 7 ⊢ 𝑡 ∈ V
23		vex 3498	. . . . . . 7 ⊢ 𝑠 ∈ V
24	22, 23	ifex 4515	. . . . . 6 ⊢ if(𝑧 = 𝑚, 𝑡, 𝑠) ∈ V
25	21, 24	opth 5361	. . . . 5 ⊢ (⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩ ↔ (𝑧 = if(𝑤 = 𝑡, 𝑛, 𝑚) ∧ if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑤))
26	20, 25	sylib 220	. . . 4 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝑧 = if(𝑤 = 𝑡, 𝑛, 𝑚) ∧ if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑤))
27	26	simprd 498	. . 3 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑤)
28		iftrue 4473	. . . . . . 7 ⊢ (𝑧 = 𝑚 → if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑡)
29	27	eqeq1d 2823	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑡 ↔ 𝑤 = 𝑡))
30	28, 29	syl5ib 246	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝑧 = 𝑚 → 𝑤 = 𝑡))
31		iftrue 4473	. . . . . . 7 ⊢ (𝑤 = 𝑡 → if(𝑤 = 𝑡, 𝑛, 𝑚) = 𝑛)
32	26	simpld 497	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑧 = if(𝑤 = 𝑡, 𝑛, 𝑚))
33	32	eqeq1d 2823	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝑧 = 𝑛 ↔ if(𝑤 = 𝑡, 𝑛, 𝑚) = 𝑛))
34	31, 33	syl5ibr 248	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝑤 = 𝑡 → 𝑧 = 𝑛))
35	30, 34	syld 47	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝑧 = 𝑚 → 𝑧 = 𝑛))
36		unxpdomlem2.2	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑚 = 𝑛)
37	36	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → ¬ 𝑚 = 𝑛)
38		equequ1 2028	. . . . . . 7 ⊢ (𝑧 = 𝑚 → (𝑧 = 𝑛 ↔ 𝑚 = 𝑛))
39	38	notbid 320	. . . . . 6 ⊢ (𝑧 = 𝑚 → (¬ 𝑧 = 𝑛 ↔ ¬ 𝑚 = 𝑛))
40	37, 39	syl5ibrcom 249	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝑧 = 𝑚 → ¬ 𝑧 = 𝑛))
41	35, 40	pm2.65d 198	. . . 4 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → ¬ 𝑧 = 𝑚)
42	41	iffalsed 4478	. . 3 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑠)
43		iffalse 4476	. . . . 5 ⊢ (¬ 𝑤 = 𝑡 → if(𝑤 = 𝑡, 𝑛, 𝑚) = 𝑚)
44	32	eqeq1d 2823	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝑧 = 𝑚 ↔ if(𝑤 = 𝑡, 𝑛, 𝑚) = 𝑚))
45	43, 44	syl5ibr 248	. . . 4 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (¬ 𝑤 = 𝑡 → 𝑧 = 𝑚))
46	41, 45	mt3d 150	. . 3 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑤 = 𝑡)
47	27, 42, 46	3eqtr3d 2864	. 2 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑠 = 𝑡)
48	2, 47	mtand 814	1 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → ¬ (𝐹‘𝑧) = (𝐹‘𝑤))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ∪ cun 3934  ifcif 4467  ⟨cop 4567   ↦ cmpt 5139  ‘cfv 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-iota 6309  df-fun 6352  df-fv 6358

This theorem is referenced by:  unxpdomlem3  8718