Theorem hsmexlem3 10338

Description: Lemma for hsmex 10342. 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 9477	. . . . 5 ⊢ (𝐶 ∈ On → 𝐶 ≼* 𝐶)
2		xpwdomg 9490	. . . . 5 ⊢ ((𝐴 ≼* 𝐷 ∧ 𝐶 ≼* 𝐶) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
3	1, 2	sylan2 593	. . . 4 ⊢ ((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
4		wdompwdom 9483	. . . 4 ⊢ ((𝐴 × 𝐶) ≼* (𝐷 × 𝐶) → 𝒫 (𝐴 × 𝐶) ≼ 𝒫 (𝐷 × 𝐶))
5		harword 9468	. . . 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 9471	. . . . . 6 ⊢ Rel ≼*
9	8	brrelex1i 5680	. . . . 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 10337	. . 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 1373	. 2 ⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
18	7, 17	sseldd 3934	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 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ⊆ wss 3901  𝒫 cpw 4554  ∪ ciun 4946   class class class wbr 5098   E cep 5523   × cxp 5622  dom cdm 5624  Oncon0 6317  ‘cfv 6492   ≼ cdom 8881  OrdIsocoi 9414  harchar 9461   ≼^* cwdom 9469

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-smo 8278  df-recs 8303  df-en 8884  df-dom 8885  df-sdom 8886  df-oi 9415  df-har 9462  df-wdom 9470

This theorem is referenced by:  hsmexlem4  10339  hsmexlem5  10340