Theorem marypha2 8905

Description: Version of marypha1 8900 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 8818	. . 3 ⊢ (𝜑 → ∪ ran 𝐹 ∈ Fin)
4		eqid 2823	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))
5	4	marypha2lem1 8901	. . . 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 6517	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
9	4	marypha2lem4 8904	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑑 ⊆ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) “ 𝑑) = ∪ (𝐹 “ 𝑑))
10	8, 9	sylan 582	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) “ 𝑑) = ∪ (𝐹 “ 𝑑))
11	7, 10	breqtrrd 5096	. . 3 ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) “ 𝑑))
12	1, 3, 6, 11	marypha1 8900	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))𝑔:𝐴–1-1→∪ ran 𝐹)
13		df-rex 3146	. . 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 3993	. . . . . . . 8 ⊢ ∪ ran 𝐹 ⊆ V
15		f1ss 6582	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1→∪ ran 𝐹 ∧ ∪ ran 𝐹 ⊆ V) → 𝑔:𝐴–1-1→V)
16	14, 15	mpan2 689	. . . . . . 7 ⊢ (𝑔:𝐴–1-1→∪ ran 𝐹 → 𝑔:𝐴–1-1→V)
17	16	ad2antll 727	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹)) → 𝑔:𝐴–1-1→V)
18		elpwi 4550	. . . . . . . 8 ⊢ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) → 𝑔 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)))
19	18	ad2antrl 726	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹)) → 𝑔 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)))
20		f1fn 6578	. . . . . . . . 9 ⊢ (𝑔:𝐴–1-1→∪ ran 𝐹 → 𝑔 Fn 𝐴)
21	20	ad2antll 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ∧ 𝑔:𝐴–1-1→∪ ran 𝐹)) → 𝑔 Fn 𝐴)
22	4	marypha2lem3 8903	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑔 Fn 𝐴) → (𝑔 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
23	8, 21, 22	syl2an2r 683	. . . . . . 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 234	. . . . . 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 514	. . . . 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 415	. . . 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	syl5bi 244	. 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 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ⊆ wss 3938  𝒫 cpw 4541  {csn 4569  ∪ cuni 4840  ∪ ciun 4921   class class class wbr 5068   × cxp 5555  ran crn 5558   “ cima 5560   Fn wfn 6352  ⟶wf 6353  –1-1→wf1 6354  ‘cfv 6357   ≼ cdom 8509  Fincfn 8511

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515

This theorem is referenced by: (None)