Theorem hsmexlem3 10497

Description: Lemma for hsmex 10501. 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 9641	. . . . 5 ⊢ (𝐶 ∈ On → 𝐶 ≼* 𝐶)
2		xpwdomg 9654	. . . . 5 ⊢ ((𝐴 ≼* 𝐷 ∧ 𝐶 ≼* 𝐶) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
3	1, 2	sylan2 592	. . . 4 ⊢ ((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
4		wdompwdom 9647	. . . 4 ⊢ ((𝐴 × 𝐶) ≼* (𝐷 × 𝐶) → 𝒫 (𝐴 × 𝐶) ≼ 𝒫 (𝐷 × 𝐶))
5		harword 9632	. . . 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 480	. 2 ⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (har‘𝒫 (𝐴 × 𝐶)) ⊆ (har‘𝒫 (𝐷 × 𝐶)))
8		relwdom 9635	. . . . . 6 ⊢ Rel ≼*
9	8	brrelex1i 5756	. . . . 5 ⊢ (𝐴 ≼* 𝐷 → 𝐴 ∈ V)
10	9	adantr 480	. . . 4 ⊢ ((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) → 𝐴 ∈ V)
11	10	adantr 480	. . 3 ⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → 𝐴 ∈ V)
12		simplr 768	. . 3 ⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → 𝐶 ∈ On)
13		simpr 484	. . 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 10496	. . 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 1371	. 2 ⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
18	7, 17	sseldd 4009	1 ⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐷 × 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622  ∪ ciun 5015   class class class wbr 5166   E cep 5598   × cxp 5698  dom cdm 5700  Oncon0 6395  ‘cfv 6573   ≼ cdom 9001  OrdIsocoi 9578  harchar 9625   ≼^* cwdom 9633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-smo 8402  df-recs 8427  df-en 9004  df-dom 9005  df-sdom 9006  df-oi 9579  df-har 9626  df-wdom 9634

This theorem is referenced by:  hsmexlem4  10498  hsmexlem5  10499