Theorem marypha1 9394

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 4574	. . . . 5 ⊢ (𝑑 ∈ 𝒫 𝐴 → 𝑑 ⊆ 𝐴)
2		marypha1.d	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (𝐶 “ 𝑑))
3	1, 2	sylan2 604	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝒫 𝐴) → 𝑑 ≼ (𝐶 “ 𝑑))
4	3	ralrimiva 3163	. . 3 ⊢ (𝜑 → ∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑))
5		imaeq1 6058	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑐 “ 𝑑) = (𝐶 “ 𝑑))
6	5	breq2d 5125	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑑 ≼ (𝑐 “ 𝑑) ↔ 𝑑 ≼ (𝐶 “ 𝑑)))
7	6	ralbidv 3194	. . . . 5 ⊢ (𝑐 = 𝐶 → (∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) ↔ ∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑)))
8		pweq 4581	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝒫 𝑐 = 𝒫 𝐶)
9	8	rexeqdv 3330	. . . . 5 ⊢ (𝑐 = 𝐶 → (∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V ↔ ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V))
10	7, 9	imbi12d 347	. . . 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 5683	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝐴 × 𝑏) = (𝐴 × 𝐵))
14	13	pweqd 4584	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → 𝒫 (𝐴 × 𝑏) = 𝒫 (𝐴 × 𝐵))
15	14	raleqdv 3329	. . . . . . 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 343	. . . . . 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 9393	. . . . . . 7 ⊢ (𝐴 ∈ Fin → (𝑏 ∈ Fin → ∀𝑐 ∈ 𝒫 (𝐴 × 𝑏)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V)))
18	17	com12 33	. . . . . 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 3550	. . . . 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 66	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ 𝒫 (𝐴 × 𝐵)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V))
21	12, 11	xpexd 7750	. . . . 5 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
22		marypha1.c	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (𝐴 × 𝐵))
23	21, 22	sselpwd 5299	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝒫 (𝐴 × 𝐵))
24	10, 20, 23	rspcdva 3591	. . 3 ⊢ (𝜑 → (∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑) → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V))
25	4, 24	mpd 16	. 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V)
26		elpwi 4574	. . . . . . 7 ⊢ (𝑓 ∈ 𝒫 𝐶 → 𝑓 ⊆ 𝐶)
27	26, 22	sylan9ssr 3959	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → 𝑓 ⊆ (𝐴 × 𝐵))
28		rnss 5930	. . . . . 6 ⊢ (𝑓 ⊆ (𝐴 × 𝐵) → ran 𝑓 ⊆ ran (𝐴 × 𝐵))
29	27, 28	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → ran 𝑓 ⊆ ran (𝐴 × 𝐵))
30		rnxpss 6171	. . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
31	29, 30	sstrdi 3957	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → ran 𝑓 ⊆ 𝐵)
32		f1ssr 6783	. . . . 5 ⊢ ((𝑓:𝐴–1-1→V ∧ ran 𝑓 ⊆ 𝐵) → 𝑓:𝐴–1-1→𝐵)
33	32	expcom 418	. . . 4 ⊢ (ran 𝑓 ⊆ 𝐵 → (𝑓:𝐴–1-1→V → 𝑓:𝐴–1-1→𝐵))
34	31, 33	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → (𝑓:𝐴–1-1→V → 𝑓:𝐴–1-1→𝐵))
35	34	reximdva 3184	. 2 ⊢ (𝜑 → (∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→𝐵))
36	25, 35	mpd 16	1 ⊢ (𝜑 → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  𝒫 cpw 4567   class class class wbr 5113   × cxp 5660  ran crn 5663   “ cima 5665  –1-1→wf1 6534   ≼ cdom 8941  Fincfn 8943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7863  df-1o 8453  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947

This theorem is referenced by:  marypha2  9399