MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dom2lem Structured version   Visualization version   GIF version

Theorem dom2lem 7946
Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.)
Hypotheses
Ref Expression
dom2d.1 (𝜑 → (𝑥𝐴𝐶𝐵))
dom2d.2 (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))
Assertion
Ref Expression
dom2lem (𝜑 → (𝑥𝐴𝐶):𝐴1-1𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem dom2lem
StepHypRef Expression
1 dom2d.1 . . . 4 (𝜑 → (𝑥𝐴𝐶𝐵))
21ralrimiv 2960 . . 3 (𝜑 → ∀𝑥𝐴 𝐶𝐵)
3 eqid 2621 . . . 4 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
43fmpt 6342 . . 3 (∀𝑥𝐴 𝐶𝐵 ↔ (𝑥𝐴𝐶):𝐴𝐵)
52, 4sylib 208 . 2 (𝜑 → (𝑥𝐴𝐶):𝐴𝐵)
61imp 445 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐶𝐵)
73fvmpt2 6253 . . . . . . . 8 ((𝑥𝐴𝐶𝐵) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
87adantll 749 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝐶𝐵) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
96, 8mpdan 701 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
109adantrr 752 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
11 nfv 1840 . . . . . . . 8 𝑥(𝜑𝑦𝐴)
12 nffvmpt1 6161 . . . . . . . . 9 𝑥((𝑥𝐴𝐶)‘𝑦)
1312nfeq1 2774 . . . . . . . 8 𝑥((𝑥𝐴𝐶)‘𝑦) = 𝐷
1411, 13nfim 1822 . . . . . . 7 𝑥((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)
15 eleq1 2686 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1615anbi2d 739 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴)))
1716imbi1d 331 . . . . . . . 8 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶) ↔ ((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)))
1815anbi1d 740 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥𝐴𝑦𝐴) ↔ (𝑦𝐴𝑦𝐴)))
19 anidm 675 . . . . . . . . . . . 12 ((𝑦𝐴𝑦𝐴) ↔ 𝑦𝐴)
2018, 19syl6bb 276 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥𝐴𝑦𝐴) ↔ 𝑦𝐴))
2120anbi2d 739 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ↔ (𝜑𝑦𝐴)))
22 fveq2 6153 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦))
2322adantr 481 . . . . . . . . . . . 12 ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥𝐴𝑦𝐴))) → ((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦))
24 dom2d.2 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))
2524imp 445 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝐶 = 𝐷𝑥 = 𝑦))
2625biimparc 504 . . . . . . . . . . . 12 ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥𝐴𝑦𝐴))) → 𝐶 = 𝐷)
2723, 26eqeq12d 2636 . . . . . . . . . . 11 ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥𝐴𝑦𝐴))) → (((𝑥𝐴𝐶)‘𝑥) = 𝐶 ↔ ((𝑥𝐴𝐶)‘𝑦) = 𝐷))
2827ex 450 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (((𝑥𝐴𝐶)‘𝑥) = 𝐶 ↔ ((𝑥𝐴𝐶)‘𝑦) = 𝐷)))
2921, 28sylbird 250 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝜑𝑦𝐴) → (((𝑥𝐴𝐶)‘𝑥) = 𝐶 ↔ ((𝑥𝐴𝐶)‘𝑦) = 𝐷)))
3029pm5.74d 262 . . . . . . . 8 (𝑥 = 𝑦 → (((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶) ↔ ((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)))
3117, 30bitrd 268 . . . . . . 7 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶) ↔ ((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)))
3214, 31, 9chvar 2261 . . . . . 6 ((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)
3332adantrl 751 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)
3410, 33eqeq12d 2636 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦) ↔ 𝐶 = 𝐷))
3525biimpd 219 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝐶 = 𝐷𝑥 = 𝑦))
3634, 35sylbid 230 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦) → 𝑥 = 𝑦))
3736ralrimivva 2966 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴 (((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦) → 𝑥 = 𝑦))
38 nfmpt1 4712 . . 3 𝑥(𝑥𝐴𝐶)
39 nfcv 2761 . . 3 𝑦(𝑥𝐴𝐶)
4038, 39dff13f 6473 . 2 ((𝑥𝐴𝐶):𝐴1-1𝐵 ↔ ((𝑥𝐴𝐶):𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦) → 𝑥 = 𝑦)))
415, 37, 40sylanbrc 697 1 (𝜑 → (𝑥𝐴𝐶):𝐴1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  cmpt 4678  wf 5848  1-1wf1 5849  cfv 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fv 5860
This theorem is referenced by:  dom2d  7947  dom3d  7948  ixpfi2  8215  infxpenc2lem1  8793  dfac12lem2  8917  4sqlem11  15590  odf1o1  17915  odf1o2  17916  dis2ndc  21182  hauspwpwf1  21710  itg1addlem4  23385  basellem3  24722  fsumvma  24851  dchrisum0fno1  25113
  Copyright terms: Public domain W3C validator