Theorem marypha1 9428

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 4609	. . . . 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 3146	. . 3 ⊢ (𝜑 → ∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑))
5		imaeq1 6054	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑐 “ 𝑑) = (𝐶 “ 𝑑))
6	5	breq2d 5160	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑑 ≼ (𝑐 “ 𝑑) ↔ 𝑑 ≼ (𝐶 “ 𝑑)))
7	6	ralbidv 3177	. . . . 5 ⊢ (𝑐 = 𝐶 → (∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) ↔ ∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑)))
8		pweq 4616	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝒫 𝑐 = 𝒫 𝐶)
9	8	rexeqdv 3326	. . . . 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 5697	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝐴 × 𝑏) = (𝐴 × 𝐵))
14	13	pweqd 4619	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → 𝒫 (𝐴 × 𝑏) = 𝒫 (𝐴 × 𝐵))
15	14	raleqdv 3325	. . . . . . 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 9427	. . . . . . 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 3565	. . . . 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 7737	. . . . 5 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
22		marypha1.c	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (𝐴 × 𝐵))
23	21, 22	sselpwd 5326	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝒫 (𝐴 × 𝐵))
24	10, 20, 23	rspcdva 3613	. . 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 4609	. . . . . . 7 ⊢ (𝑓 ∈ 𝒫 𝐶 → 𝑓 ⊆ 𝐶)
27	26, 22	sylan9ssr 3996	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → 𝑓 ⊆ (𝐴 × 𝐵))
28		rnss 5938	. . . . . 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 6171	. . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
31	29, 30	sstrdi 3994	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → ran 𝑓 ⊆ 𝐵)
32		f1ssr 6794	. . . . 5 ⊢ ((𝑓:𝐴–1-1→V ∧ ran 𝑓 ⊆ 𝐵) → 𝑓:𝐴–1-1→𝐵)
33	32	expcom 414	. . . 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 3168	. 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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ⊆ wss 3948  𝒫 cpw 4602   class class class wbr 5148   × cxp 5674  ran crn 5677   “ cima 5679  –1-1→wf1 6540   ≼ cdom 8936  Fincfn 8938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7855  df-1o 8465  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942

This theorem is referenced by:  marypha2  9433