Theorem marypha2 9340

Description: Version of marypha1 9335 using a functional family of sets instead of a relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Hypotheses

Ref	Expression
marypha2.a	⊢ (𝜑 → 𝐴 ∈ Fin)
marypha2.b	⊢ (𝜑 → 𝐹:𝐴⟶Fin)
marypha2.c	⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ ∪ (𝐹 “ 𝑑))

Assertion

Ref	Expression
marypha2	⊢ (𝜑 → ∃𝑔(𝑔:𝐴–1-1→V ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))

Distinct variable groups:   𝜑,𝑑,𝑔,𝑥   𝐴,𝑑,𝑔,𝑥   𝐹,𝑑,𝑔,𝑥

Proof of Theorem marypha2

Step	Hyp	Ref	Expression
1		marypha2.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
2		marypha2.b	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶Fin)
3	2, 1	unirnffid 9245	. . 3 ⊢ (𝜑 → ∪ ran 𝐹 ∈ Fin)
4		eqid 2734	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))
5	4	marypha2lem1 9336	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹)
6	5	a1i 11	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹))
7		marypha2.c	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ ∪ (𝐹 “ 𝑑))
8	2	ffnd 6661	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
9	4	marypha2lem4 9339	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑑 ⊆ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) “ 𝑑) = ∪ (𝐹 “ 𝑑))
10	8, 9	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) “ 𝑑) = ∪ (𝐹 “ 𝑑))
11	7, 10	breqtrrd 5124	. . 3 ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) “ 𝑑))
12	1, 3, 6, 11	marypha1 9335	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))𝑔:𝐴–1-1→∪ ran 𝐹)
13		df-rex 3059	. . 3 ⊢ (∃𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))𝑔:𝐴–1-1→∪ ran 𝐹 ↔ ∃𝑔(𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹))
14		ssv 3956	. . . . . . . 8 ⊢ ∪ ran 𝐹 ⊆ V
15		f1ss 6733	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1→∪ ran 𝐹 ∧ ∪ ran 𝐹 ⊆ V) → 𝑔:𝐴–1-1→V)
16	14, 15	mpan2 691	. . . . . . 7 ⊢ (𝑔:𝐴–1-1→∪ ran 𝐹 → 𝑔:𝐴–1-1→V)
17	16	ad2antll 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹)) → 𝑔:𝐴–1-1→V)
18		elpwi 4559	. . . . . . . 8 ⊢ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) → 𝑔 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)))
19	18	ad2antrl 728	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹)) → 𝑔 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)))
20		f1fn 6729	. . . . . . . . 9 ⊢ (𝑔:𝐴–1-1→∪ ran 𝐹 → 𝑔 Fn 𝐴)
21	20	ad2antll 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹)) → 𝑔 Fn 𝐴)
22	4	marypha2lem3 9338	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑔 Fn 𝐴) → (𝑔 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
23	8, 21, 22	syl2an2r 685	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹)) → (𝑔 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
24	19, 23	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹)) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥))
25	17, 24	jca 511	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹)) → (𝑔:𝐴–1-1→V ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
26	25	ex 412	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹) → (𝑔:𝐴–1-1→V ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥))))
27	26	eximdv 1918	. . 3 ⊢ (𝜑 → (∃𝑔(𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹) → ∃𝑔(𝑔:𝐴–1-1→V ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥))))
28	13, 27	biimtrid 242	. 2 ⊢ (𝜑 → (∃𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))𝑔:𝐴–1-1→∪ ran 𝐹 → ∃𝑔(𝑔:𝐴–1-1→V ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥))))
29	12, 28	mpd 15	1 ⊢ (𝜑 → ∃𝑔(𝑔:𝐴–1-1→V ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  Vcvv 3438   ⊆ wss 3899  𝒫 cpw 4552  {csn 4578  ∪ cuni 4861  ∪ ciun 4944   class class class wbr 5096   × cxp 5620  ran crn 5623   “ cima 5625   Fn wfn 6485  ⟶wf 6486  –1-1→wf1 6487  ‘cfv 6490   ≼ cdom 8879  Fincfn 8881

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 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-om 7807  df-1st 7931  df-2nd 7932  df-1o 8395  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885

This theorem is referenced by: (None)