Theorem marypha2 9406

Description: Version of marypha1 9401 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 9311	. . 3 ⊢ (𝜑 → ∪ ran 𝐹 ∈ Fin)
4		eqid 2765	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))
5	4	marypha2lem1 9402	. . . 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 6710	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
9	4	marypha2lem4 9405	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑑 ⊆ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) “ 𝑑) = ∪ (𝐹 “ 𝑑))
10	8, 9	sylan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) “ 𝑑) = ∪ (𝐹 “ 𝑑))
11	7, 10	breqtrrd 5141	. . 3 ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) “ 𝑑))
12	1, 3, 6, 11	marypha1 9401	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))𝑔:𝐴–1-1→∪ ran 𝐹)
13		df-rex 3092	. . 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 3962	. . . . . . . 8 ⊢ ∪ ran 𝐹 ⊆ V
15		f1ss 6785	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1→∪ ran 𝐹 ∧ ∪ ran 𝐹 ⊆ V) → 𝑔:𝐴–1-1→V)
16	14, 15	mpan2 704	. . . . . . 7 ⊢ (𝑔:𝐴–1-1→∪ ran 𝐹 → 𝑔:𝐴–1-1→V)
17	16	ad2antll 742	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹)) → 𝑔:𝐴–1-1→V)
18		elpwi 4571	. . . . . . . 8 ⊢ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) → 𝑔 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)))
19	18	ad2antrl 741	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹)) → 𝑔 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)))
20		f1fn 6779	. . . . . . . . 9 ⊢ (𝑔:𝐴–1-1→∪ ran 𝐹 → 𝑔 Fn 𝐴)
21	20	ad2antll 742	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹)) → 𝑔 Fn 𝐴)
22	4	marypha2lem3 9404	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑔 Fn 𝐴) → (𝑔 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
23	8, 21, 22	syl2an2r 698	. . . . . . 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 235	. . . . . 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 521	. . . . 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 418	. . . 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 1950	. . 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 245	. 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 16	1 ⊢ (𝜑 → ∃𝑔(𝑔:𝐴–1-1→V ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  𝒫 cpw 4564  {csn 4591  ∪ cuni 4874  ∪ ciun 4958   class class class wbr 5111   × cxp 5661  ran crn 5664   “ cima 5666   Fn wfn 6535  ⟶wf 6536  –1-1→wf1 6537  ‘cfv 6540   ≼ cdom 8947  Fincfn 8949

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7869  df-1st 7992  df-2nd 7993  df-1o 8459  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953

This theorem is used by: (None)