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Theorem wdomtr 9593

Description: Transitivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)

**Assertion**
Ref	Expression
wdomtr	⊢ ((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) → 𝑋 ≼^* 𝑍)

Proof of Theorem wdomtr Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relwdom 9584	. . . . 5 ⊢ Rel ≼^*
2	1	brrelex2i 5712	. . . 4 ⊢ (𝑌 ≼^* 𝑍 → 𝑍 ∈ V)
3	2	adantl 487	. . 3 ⊢ ((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) → 𝑍 ∈ V)
4		0wdom 9588	. . . 4 ⊢ (𝑍 ∈ V → ∅ ≼^* 𝑍)
5		breq1 5109	. . . 4 ⊢ (𝑋 = ∅ → (𝑋 ≼^* 𝑍 ↔ ∅ ≼^* 𝑍))
6	4, 5	syl5ibrcom 250	. . 3 ⊢ (𝑍 ∈ V → (𝑋 = ∅ → 𝑋 ≼^* 𝑍))
7	3, 6	syl 18	. 2 ⊢ ((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) → (𝑋 = ∅ → 𝑋 ≼^* 𝑍))
8		simpll 779	. . . . 5 ⊢ (((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) → 𝑋 ≼^* 𝑌)
9		brwdomn0 9587	. . . . . 6 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌⟶onto𝑋))
10	9	adantl 487	. . . . 5 ⊢ (((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌⟶onto𝑋))
11	8, 10	mpbid 235	. . . 4 ⊢ (((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) → ∃𝑧 𝑧:𝑌⟶onto𝑋)
12		simpllr 788	. . . . . 6 ⊢ ((((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) ∧ 𝑧:𝑌⟶onto𝑋) → 𝑌 ≼^* 𝑍)
13		simplr 781	. . . . . . . 8 ⊢ ((((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) ∧ 𝑧:𝑌⟶onto𝑋) → 𝑋 ≠ ∅)
14		dm0rn0 5912	. . . . . . . . . . . 12 ⊢ (dom 𝑧 = ∅ ↔ ran 𝑧 = ∅)
15	14	necon3bii 3011	. . . . . . . . . . 11 ⊢ (dom 𝑧 ≠ ∅ ↔ ran 𝑧 ≠ ∅)
16	15	a1i 11	. . . . . . . . . 10 ⊢ (𝑧:𝑌⟶onto𝑋 → (dom 𝑧 ≠ ∅ ↔ ran 𝑧 ≠ ∅))
17		fof 6815	. . . . . . . . . . . 12 ⊢ (𝑧:𝑌⟶onto𝑋 → 𝑧:𝑌⟶𝑋)
18	17	fdmd 6739	. . . . . . . . . . 11 ⊢ (𝑧:𝑌⟶onto𝑋 → dom 𝑧 = 𝑌)
19	18	neeq1d 3018	. . . . . . . . . 10 ⊢ (𝑧:𝑌⟶onto𝑋 → (dom 𝑧 ≠ ∅ ↔ 𝑌 ≠ ∅))
20		forn 6818	. . . . . . . . . . 11 ⊢ (𝑧:𝑌⟶onto𝑋 → ran 𝑧 = 𝑋)
21	20	neeq1d 3018	. . . . . . . . . 10 ⊢ (𝑧:𝑌⟶onto𝑋 → (ran 𝑧 ≠ ∅ ↔ 𝑋 ≠ ∅))
22	16, 19, 21	3bitr3rd 313	. . . . . . . . 9 ⊢ (𝑧:𝑌⟶onto𝑋 → (𝑋 ≠ ∅ ↔ 𝑌 ≠ ∅))
23	22	adantl 487	. . . . . . . 8 ⊢ ((((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) ∧ 𝑧:𝑌⟶onto𝑋) → (𝑋 ≠ ∅ ↔ 𝑌 ≠ ∅))
24	13, 23	mpbid 235	. . . . . . 7 ⊢ ((((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) ∧ 𝑧:𝑌⟶onto𝑋) → 𝑌 ≠ ∅)
25		brwdomn0 9587	. . . . . . 7 ⊢ (𝑌 ≠ ∅ → (𝑌 ≼^* 𝑍 ↔ ∃𝑦 𝑦:𝑍⟶onto𝑌))
26	24, 25	syl 18	. . . . . 6 ⊢ ((((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) ∧ 𝑧:𝑌⟶onto𝑋) → (𝑌 ≼^* 𝑍 ↔ ∃𝑦 𝑦:𝑍⟶onto𝑌))
27	12, 26	mpbid 235	. . . . 5 ⊢ ((((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) ∧ 𝑧:𝑌⟶onto𝑋) → ∃𝑦 𝑦:𝑍⟶onto𝑌)
28		vex 3458	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
29		vex 3458	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
30	28, 29	coex 7963	. . . . . . . . 9 ⊢ (𝑧 ∘ 𝑦) ∈ V
31		foco 6829	. . . . . . . . 9 ⊢ ((𝑧:𝑌⟶onto𝑋 ∧ 𝑦:𝑍⟶onto𝑌) → (𝑧 ∘ 𝑦):𝑍⟶onto𝑋)
32		fowdom 9589	. . . . . . . . 9 ⊢ (((𝑧 ∘ 𝑦) ∈ V ∧ (𝑧 ∘ 𝑦):𝑍⟶onto𝑋) → 𝑋 ≼^* 𝑍)
33	30, 31, 32	sylancr 599	. . . . . . . 8 ⊢ ((𝑧:𝑌⟶onto𝑋 ∧ 𝑦:𝑍⟶onto𝑌) → 𝑋 ≼^* 𝑍)
34	33	adantl 487	. . . . . . 7 ⊢ ((((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) ∧ (𝑧:𝑌⟶onto𝑋 ∧ 𝑦:𝑍⟶onto𝑌)) → 𝑋 ≼^* 𝑍)
35	34	expr 462	. . . . . 6 ⊢ ((((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) ∧ 𝑧:𝑌⟶onto𝑋) → (𝑦:𝑍⟶onto𝑌 → 𝑋 ≼^* 𝑍))
36	35	exlimdv 1968	. . . . 5 ⊢ ((((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) ∧ 𝑧:𝑌⟶onto𝑋) → (∃𝑦 𝑦:𝑍⟶onto𝑌 → 𝑋 ≼^* 𝑍))
37	27, 36	mpd 16	. . . 4 ⊢ ((((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) ∧ 𝑧:𝑌⟶onto𝑋) → 𝑋 ≼^* 𝑍)
38	11, 37	exlimddv 1970	. . 3 ⊢ (((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) ∧ 𝑋 ≠ ∅) → 𝑋 ≼^* 𝑍)
39	38	ex 418	. 2 ⊢ ((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) → (𝑋 ≠ ∅ → 𝑋 ≼^* 𝑍))
40	7, 39	pm2.61dne 3045	1 ⊢ ((𝑋 ≼^* 𝑌 ∧ 𝑌 ≼^* 𝑍) → 𝑋 ≼^* 𝑍)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2147 ≠ wne 2959 Vcvv 3454 ∅c0 4281 class class class wbr 5106 dom cdm 5655 ran crn 5656 ∘ ccom 5659 ⟶ontowfo 6553 ≼^* cwdom 9582

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5171 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-fun 6557 df-fn 6558 df-f 6559 df-fo 6561 df-wdom 9583

This theorem is used by: wdomen1 9594 wdomen2 9595 wdom2d 9598 wdomima2g 9604 unxpwdom2 9606 unxpwdom 9607 harwdom 9609 pwdjudom 10317 hsmexlem1 10528 hsmexlem4 10531

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