Theorem marypha1 8223

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 4117	. . . . 5 ⊢ (𝑑 ∈ 𝒫 𝐴 → 𝑑 ⊆ 𝐴)
2		marypha1.d	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (𝐶 “ 𝑑))
3	1, 2	sylan2 490	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝒫 𝐴) → 𝑑 ≼ (𝐶 “ 𝑑))
4	3	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑))
5		marypha1.c	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (𝐴 × 𝐵))
6		marypha1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Fin)
7		marypha1.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin)
8		xpexg 6858	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ V)
9	6, 7, 8	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
10		elpw2g 4754	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ V → (𝐶 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝐶 ⊆ (𝐴 × 𝐵)))
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝐶 ⊆ (𝐴 × 𝐵)))
12	5, 11	mpbird 246	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝒫 (𝐴 × 𝐵))
13		xpeq2 5053	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝐴 × 𝑏) = (𝐴 × 𝐵))
14	13	pweqd 4113	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → 𝒫 (𝐴 × 𝑏) = 𝒫 (𝐴 × 𝐵))
15	14	raleqdv 3121	. . . . . . 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 329	. . . . . 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 8222	. . . . . . 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 3245	. . . . 5 ⊢ (𝐵 ∈ Fin → (𝐴 ∈ Fin → ∀𝑐 ∈ 𝒫 (𝐴 × 𝐵)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V)))
20	7, 6, 19	sylc 63	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ 𝒫 (𝐴 × 𝐵)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V))
21		imaeq1 5380	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (𝑐 “ 𝑑) = (𝐶 “ 𝑑))
22	21	breq2d 4595	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑑 ≼ (𝑐 “ 𝑑) ↔ 𝑑 ≼ (𝐶 “ 𝑑)))
23	22	ralbidv 2969	. . . . . 6 ⊢ (𝑐 = 𝐶 → (∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) ↔ ∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑)))
24		pweq 4111	. . . . . . 7 ⊢ (𝑐 = 𝐶 → 𝒫 𝑐 = 𝒫 𝐶)
25	24	rexeqdv 3122	. . . . . 6 ⊢ (𝑐 = 𝐶 → (∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V ↔ ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V))
26	23, 25	imbi12d 333	. . . . 5 ⊢ (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)))
27	26	rspcva 3280	. . . 4 ⊢ ((𝐶 ∈ 𝒫 (𝐴 × 𝐵) ∧ ∀𝑐 ∈ 𝒫 (𝐴 × 𝐵)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V)) → (∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V))
28	12, 20, 27	syl2anc 691	. . 3 ⊢ (𝜑 → (∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V))
29	4, 28	mpd 15	. 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V)
30		elpwi 4117	. . . . . . 7 ⊢ (𝑓 ∈ 𝒫 𝐶 → 𝑓 ⊆ 𝐶)
31	30, 5	sylan9ssr 3582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → 𝑓 ⊆ (𝐴 × 𝐵))
32		rnss 5275	. . . . . 6 ⊢ (𝑓 ⊆ (𝐴 × 𝐵) → ran 𝑓 ⊆ ran (𝐴 × 𝐵))
33	31, 32	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → ran 𝑓 ⊆ ran (𝐴 × 𝐵))
34		rnxpss 5485	. . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
35	33, 34	syl6ss 3580	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → ran 𝑓 ⊆ 𝐵)
36		f1ssr 6020	. . . . 5 ⊢ ((𝑓:𝐴–1-1→V ∧ ran 𝑓 ⊆ 𝐵) → 𝑓:𝐴–1-1→𝐵)
37	36	expcom 450	. . . 4 ⊢ (ran 𝑓 ⊆ 𝐵 → (𝑓:𝐴–1-1→V → 𝑓:𝐴–1-1→𝐵))
38	35, 37	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → (𝑓:𝐴–1-1→V → 𝑓:𝐴–1-1→𝐵))
39	38	reximdva 3000	. 2 ⊢ (𝜑 → (∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→𝐵))
40	29, 39	mpd 15	1 ⊢ (𝜑 → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583   × cxp 5036  ran crn 5039   “ cima 5041  –1-1→wf1 5801   ≼ cdom 7839  Fincfn 7841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845

This theorem is referenced by:  marypha2  8228