Theorem marypha1 9324

Description: (Philip) Hall's marriage theorem, sufficiency: a finite relation contains an injection if there is no subset of its domain which would be forced to violate the pigeonhole principle. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Hypotheses

Ref	Expression
marypha1.a	⊢ (𝜑 → 𝐴 ∈ Fin)
marypha1.b	⊢ (𝜑 → 𝐵 ∈ Fin)
marypha1.c	⊢ (𝜑 → 𝐶 ⊆ (𝐴 × 𝐵))
marypha1.d	⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (𝐶 “ 𝑑))

Assertion

Ref	Expression
marypha1	⊢ (𝜑 → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→𝐵)

Distinct variable groups:   𝜑,𝑑,𝑓   𝐴,𝑑,𝑓   𝐶,𝑑,𝑓

Allowed substitution hints:   𝐵(𝑓,𝑑)

Proof of Theorem marypha1

Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4558	. . . . 5 ⊢ (𝑑 ∈ 𝒫 𝐴 → 𝑑 ⊆ 𝐴)
2		marypha1.d	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (𝐶 “ 𝑑))
3	1, 2	sylan2 593	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝒫 𝐴) → 𝑑 ≼ (𝐶 “ 𝑑))
4	3	ralrimiva 3121	. . 3 ⊢ (𝜑 → ∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑))
5		imaeq1 6006	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑐 “ 𝑑) = (𝐶 “ 𝑑))
6	5	breq2d 5104	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑑 ≼ (𝑐 “ 𝑑) ↔ 𝑑 ≼ (𝐶 “ 𝑑)))
7	6	ralbidv 3152	. . . . 5 ⊢ (𝑐 = 𝐶 → (∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) ↔ ∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑)))
8		pweq 4565	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝒫 𝑐 = 𝒫 𝐶)
9	8	rexeqdv 3290	. . . . 5 ⊢ (𝑐 = 𝐶 → (∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V ↔ ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V))
10	7, 9	imbi12d 344	. . . 4 ⊢ (𝑐 = 𝐶 → ((∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V) ↔ (∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V)))
11		marypha1.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ Fin)
12		marypha1.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
13		xpeq2 5640	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝐴 × 𝑏) = (𝐴 × 𝐵))
14	13	pweqd 4568	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → 𝒫 (𝐴 × 𝑏) = 𝒫 (𝐴 × 𝐵))
15	14	raleqdv 3289	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑐 ∈ 𝒫 (𝐴 × 𝑏)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V) ↔ ∀𝑐 ∈ 𝒫 (𝐴 × 𝐵)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V)))
16	15	imbi2d 340	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ Fin → ∀𝑐 ∈ 𝒫 (𝐴 × 𝑏)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V)) ↔ (𝐴 ∈ Fin → ∀𝑐 ∈ 𝒫 (𝐴 × 𝐵)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V))))
17		marypha1lem 9323	. . . . . . 7 ⊢ (𝐴 ∈ Fin → (𝑏 ∈ Fin → ∀𝑐 ∈ 𝒫 (𝐴 × 𝑏)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V)))
18	17	com12 32	. . . . . 6 ⊢ (𝑏 ∈ Fin → (𝐴 ∈ Fin → ∀𝑐 ∈ 𝒫 (𝐴 × 𝑏)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V)))
19	16, 18	vtoclga 3532	. . . . 5 ⊢ (𝐵 ∈ Fin → (𝐴 ∈ Fin → ∀𝑐 ∈ 𝒫 (𝐴 × 𝐵)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V)))
20	11, 12, 19	sylc 65	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ 𝒫 (𝐴 × 𝐵)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V))
21	12, 11	xpexd 7687	. . . . 5 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
22		marypha1.c	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (𝐴 × 𝐵))
23	21, 22	sselpwd 5267	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝒫 (𝐴 × 𝐵))
24	10, 20, 23	rspcdva 3578	. . 3 ⊢ (𝜑 → (∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V))
25	4, 24	mpd 15	. 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V)
26		elpwi 4558	. . . . . . 7 ⊢ (𝑓 ∈ 𝒫 𝐶 → 𝑓 ⊆ 𝐶)
27	26, 22	sylan9ssr 3950	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → 𝑓 ⊆ (𝐴 × 𝐵))
28		rnss 5881	. . . . . 6 ⊢ (𝑓 ⊆ (𝐴 × 𝐵) → ran 𝑓 ⊆ ran (𝐴 × 𝐵))
29	27, 28	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → ran 𝑓 ⊆ ran (𝐴 × 𝐵))
30		rnxpss 6121	. . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
31	29, 30	sstrdi 3948	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → ran 𝑓 ⊆ 𝐵)
32		f1ssr 6726	. . . . 5 ⊢ ((𝑓:𝐴–1-1→V ∧ ran 𝑓 ⊆ 𝐵) → 𝑓:𝐴–1-1→𝐵)
33	32	expcom 413	. . . 4 ⊢ (ran 𝑓 ⊆ 𝐵 → (𝑓:𝐴–1-1→V → 𝑓:𝐴–1-1→𝐵))
34	31, 33	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → (𝑓:𝐴–1-1→V → 𝑓:𝐴–1-1→𝐵))
35	34	reximdva 3142	. 2 ⊢ (𝜑 → (∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→𝐵))
36	25, 35	mpd 15	1 ⊢ (𝜑 → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ⊆ wss 3903  𝒫 cpw 4551   class class class wbr 5092   × cxp 5617  ran crn 5620   “ cima 5622  –1-1→wf1 6479   ≼ cdom 8870  Fincfn 8872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-om 7800  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876

This theorem is referenced by:  marypha2  9329