Theorem marypha1 8685

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 4426	. . . . 5 ⊢ (𝑑 ∈ 𝒫 𝐴 → 𝑑 ⊆ 𝐴)
2		marypha1.d	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (𝐶 “ 𝑑))
3	1, 2	sylan2 583	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝒫 𝐴) → 𝑑 ≼ (𝐶 “ 𝑑))
4	3	ralrimiva 3126	. . 3 ⊢ (𝜑 → ∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑))
5		imaeq1 5759	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑐 “ 𝑑) = (𝐶 “ 𝑑))
6	5	breq2d 4935	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑑 ≼ (𝑐 “ 𝑑) ↔ 𝑑 ≼ (𝐶 “ 𝑑)))
7	6	ralbidv 3141	. . . . 5 ⊢ (𝑐 = 𝐶 → (∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) ↔ ∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝐶 “ 𝑑)))
8		pweq 4419	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝒫 𝑐 = 𝒫 𝐶)
9	8	rexeqdv 3350	. . . . 5 ⊢ (𝑐 = 𝐶 → (∃𝑓 ∈ 𝒫 𝑐𝑓:𝐴–1-1→V ↔ ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→V))
10	7, 9	imbi12d 337	. . . 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 5421	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝐴 × 𝑏) = (𝐴 × 𝐵))
14	13	pweqd 4421	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → 𝒫 (𝐴 × 𝑏) = 𝒫 (𝐴 × 𝐵))
15	14	raleqdv 3349	. . . . . . 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 333	. . . . . 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 8684	. . . . . . 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 3487	. . . . 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 7285	. . . . 5 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
22		marypha1.c	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (𝐴 × 𝐵))
23	21, 22	sselpwd 5080	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝒫 (𝐴 × 𝐵))
24	10, 20, 23	rspcdva 3535	. . 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 4426	. . . . . . 7 ⊢ (𝑓 ∈ 𝒫 𝐶 → 𝑓 ⊆ 𝐶)
27	26, 22	sylan9ssr 3868	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → 𝑓 ⊆ (𝐴 × 𝐵))
28		rnss 5645	. . . . . 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 5863	. . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
31	29, 30	syl6ss 3866	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝒫 𝐶) → ran 𝑓 ⊆ 𝐵)
32		f1ssr 6404	. . . . 5 ⊢ ((𝑓:𝐴–1-1→V ∧ ran 𝑓 ⊆ 𝐵) → 𝑓:𝐴–1-1→𝐵)
33	32	expcom 406	. . . 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 3213	. 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 387   = wceq 1507   ∈ wcel 2048  ∀wral 3082  ∃wrex 3083  Vcvv 3409   ⊆ wss 3825  𝒫 cpw 4416   class class class wbr 4923   × cxp 5398  ran crn 5401   “ cima 5403  –1-1→wf1 6179   ≼ cdom 8296  Fincfn 8298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-om 7391  df-1o 7897  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302

This theorem is referenced by:  marypha2  8690