Theorem hsmexlem3 9250

Description: Lemma for hsmex 9254. Clear 𝐼 hypothesis and extend previous result by dominance. Note that this could be substantially strengthened, e.g. using the weak Hartogs function, but all we need here is that there be *some* dominating ordinal. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
hsmexlem.f	⊢ 𝐹 = OrdIso( E , 𝐵)
hsmexlem.g	⊢ 𝐺 = OrdIso( E , ∪ 𝑎 ∈ 𝐴 𝐵)

Assertion

Ref	Expression
hsmexlem3	⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐷 × 𝐶)))

Distinct variable groups:   𝐴,𝑎   𝐶,𝑎

Allowed substitution hints:   𝐵(𝑎)   𝐷(𝑎)   𝐹(𝑎)   𝐺(𝑎)

Proof of Theorem hsmexlem3

Step	Hyp	Ref	Expression
1		wdomref 8477	. . . . 5 ⊢ (𝐶 ∈ On → 𝐶 ≼* 𝐶)
2		xpwdomg 8490	. . . . 5 ⊢ ((𝐴 ≼* 𝐷 ∧ 𝐶 ≼* 𝐶) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
3	1, 2	sylan2 491	. . . 4 ⊢ ((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
4		wdompwdom 8483	. . . 4 ⊢ ((𝐴 × 𝐶) ≼* (𝐷 × 𝐶) → 𝒫 (𝐴 × 𝐶) ≼ 𝒫 (𝐷 × 𝐶))
5		harword 8470	. . . 4 ⊢ (𝒫 (𝐴 × 𝐶) ≼ 𝒫 (𝐷 × 𝐶) → (har‘𝒫 (𝐴 × 𝐶)) ⊆ (har‘𝒫 (𝐷 × 𝐶)))
6	3, 4, 5	3syl 18	. . 3 ⊢ ((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) → (har‘𝒫 (𝐴 × 𝐶)) ⊆ (har‘𝒫 (𝐷 × 𝐶)))
7	6	adantr 481	. 2 ⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (har‘𝒫 (𝐴 × 𝐶)) ⊆ (har‘𝒫 (𝐷 × 𝐶)))
8		relwdom 8471	. . . . . 6 ⊢ Rel ≼*
9	8	brrelexi 5158	. . . . 5 ⊢ (𝐴 ≼* 𝐷 → 𝐴 ∈ V)
10	9	adantr 481	. . . 4 ⊢ ((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) → 𝐴 ∈ V)
11	10	adantr 481	. . 3 ⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → 𝐴 ∈ V)
12		simplr 792	. . 3 ⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → 𝐶 ∈ On)
13		simpr 477	. . 3 ⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶))
14		hsmexlem.f	. . . 4 ⊢ 𝐹 = OrdIso( E , 𝐵)
15		hsmexlem.g	. . . 4 ⊢ 𝐺 = OrdIso( E , ∪ 𝑎 ∈ 𝐴 𝐵)
16	14, 15	hsmexlem2 9249	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
17	11, 12, 13, 16	syl3anc 1326	. 2 ⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
18	7, 17	sseldd 3604	1 ⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐷 × 𝐶)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158  ∪ ciun 4520   class class class wbr 4653   E cep 5028   × cxp 5112  dom cdm 5114  Oncon0 5723  ‘cfv 5888   ≼ cdom 7953  OrdIsocoi 8414  harchar 8461   ≼^* cwdom 8462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-smo 7443  df-recs 7468  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-oi 8415  df-har 8463  df-wdom 8464

This theorem is referenced by:  hsmexlem4  9251  hsmexlem5  9252