Theorem marypha2 9342

Description: Version of marypha1 9337 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 9247	. . 3 ⊢ (𝜑 → ∪ ran 𝐹 ∈ Fin)
4		eqid 2736	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))
5	4	marypha2lem1 9338	. . . 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 6663	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
9	4	marypha2lem4 9341	. . . . 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 5126	. . 3 ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) “ 𝑑))
12	1, 3, 6, 11	marypha1 9337	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))𝑔:𝐴–1-1→∪ ran 𝐹)
13		df-rex 3061	. . 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 3958	. . . . . . . 8 ⊢ ∪ ran 𝐹 ⊆ V
15		f1ss 6735	. . . . . . . 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 4561	. . . . . . . 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 6731	. . . . . . . . 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 9340	. . . . . . . 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 3051  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901  𝒫 cpw 4554  {csn 4580  ∪ cuni 4863  ∪ ciun 4946   class class class wbr 5098   × cxp 5622  ran crn 5625   “ cima 5627   Fn wfn 6487  ⟶wf 6488  –1-1→wf1 6489  ‘cfv 6492   ≼ cdom 8881  Fincfn 8883

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-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-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-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-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-om 7809  df-1st 7933  df-2nd 7934  df-1o 8397  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887

This theorem is referenced by: (None)