Theorem marypha2 9477

Description: Version of marypha1 9472 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 9385	. . 3 ⊢ (𝜑 → ∪ ran 𝐹 ∈ Fin)
4		eqid 2735	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))
5	4	marypha2lem1 9473	. . . 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 6738	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
9	4	marypha2lem4 9476	. . . . 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 5176	. . 3 ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) “ 𝑑))
12	1, 3, 6, 11	marypha1 9472	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝒫 ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))𝑔:𝐴–1-1→∪ ran 𝐹)
13		df-rex 3069	. . 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 4020	. . . . . . . 8 ⊢ ∪ ran 𝐹 ⊆ V
15		f1ss 6810	. . . . . . . 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 4612	. . . . . . . 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 6806	. . . . . . . . 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 9475	. . . . . . . 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 1915	. . 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 1537  ∃wex 1776   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605  {csn 4631  ∪ cuni 4912  ∪ ciun 4996   class class class wbr 5148   × cxp 5687  ran crn 5690   “ cima 5692   Fn wfn 6558  ⟶wf 6559  –1-1→wf1 6560  ‘cfv 6563   ≼ cdom 8982  Fincfn 8984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988

This theorem is referenced by: (None)