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Theorem fodomfir 9368
Description: There exists a mapping from a finite set onto any nonempty set that it dominates, proved without using the Axiom of Power Sets (unlike fodomr 9168). (Contributed by BTernaryTau, 23-Jun-2025.)
Assertion
Ref Expression
fodomfir ((𝐴 ∈ Fin ∧ ∅ ≺ 𝐵𝐵𝐴) → ∃𝑓 𝑓:𝐴onto𝐵)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem fodomfir
Dummy variables 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relsdom 8992 . . . . . . 7 Rel ≺
21brrelex2i 5742 . . . . . 6 (∅ ≺ 𝐵𝐵 ∈ V)
3 0sdomg 9144 . . . . . . 7 (𝐵 ∈ V → (∅ ≺ 𝐵𝐵 ≠ ∅))
4 n0 4353 . . . . . . 7 (𝐵 ≠ ∅ ↔ ∃𝑧 𝑧𝐵)
53, 4bitrdi 287 . . . . . 6 (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧𝐵))
62, 5syl 17 . . . . 5 (∅ ≺ 𝐵 → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧𝐵))
76ibi 267 . . . 4 (∅ ≺ 𝐵 → ∃𝑧 𝑧𝐵)
8 domfi 9229 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐵 ∈ Fin)
9 simpl 482 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐴 ∈ Fin)
10 brdomi 8999 . . . . . . . 8 (𝐵𝐴 → ∃𝑔 𝑔:𝐵1-1𝐴)
11 f1fn 6805 . . . . . . . . . . . 12 (𝑔:𝐵1-1𝐴𝑔 Fn 𝐵)
12 fnfi 9218 . . . . . . . . . . . 12 ((𝑔 Fn 𝐵𝐵 ∈ Fin) → 𝑔 ∈ Fin)
1311, 12sylan 580 . . . . . . . . . . 11 ((𝑔:𝐵1-1𝐴𝐵 ∈ Fin) → 𝑔 ∈ Fin)
1413ex 412 . . . . . . . . . 10 (𝑔:𝐵1-1𝐴 → (𝐵 ∈ Fin → 𝑔 ∈ Fin))
15 cnvfi 9216 . . . . . . . . . . . . . 14 (𝑔 ∈ Fin → 𝑔 ∈ Fin)
16 diffi 9215 . . . . . . . . . . . . . . 15 (𝐴 ∈ Fin → (𝐴 ∖ ran 𝑔) ∈ Fin)
17 snfi 9083 . . . . . . . . . . . . . . 15 {𝑧} ∈ Fin
18 xpfi 9358 . . . . . . . . . . . . . . 15 (((𝐴 ∖ ran 𝑔) ∈ Fin ∧ {𝑧} ∈ Fin) → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ Fin)
1916, 17, 18sylancl 586 . . . . . . . . . . . . . 14 (𝐴 ∈ Fin → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ Fin)
20 unfi 9211 . . . . . . . . . . . . . 14 ((𝑔 ∈ Fin ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ Fin) → (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ Fin)
2115, 19, 20syl2an 596 . . . . . . . . . . . . 13 ((𝑔 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ Fin)
22 df-f1 6566 . . . . . . . . . . . . . . . . . . 19 (𝑔:𝐵1-1𝐴 ↔ (𝑔:𝐵𝐴 ∧ Fun 𝑔))
2322simprbi 496 . . . . . . . . . . . . . . . . . 18 (𝑔:𝐵1-1𝐴 → Fun 𝑔)
24 vex 3484 . . . . . . . . . . . . . . . . . . . 20 𝑧 ∈ V
2524fconst 6794 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧}
26 ffun 6739 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧} → Fun ((𝐴 ∖ ran 𝑔) × {𝑧}))
2725, 26ax-mp 5 . . . . . . . . . . . . . . . . . 18 Fun ((𝐴 ∖ ran 𝑔) × {𝑧})
2823, 27jctir 520 . . . . . . . . . . . . . . . . 17 (𝑔:𝐵1-1𝐴 → (Fun 𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})))
29 df-rn 5696 . . . . . . . . . . . . . . . . . . . 20 ran 𝑔 = dom 𝑔
3029eqcomi 2746 . . . . . . . . . . . . . . . . . . 19 dom 𝑔 = ran 𝑔
3124snnz 4776 . . . . . . . . . . . . . . . . . . . 20 {𝑧} ≠ ∅
32 dmxp 5939 . . . . . . . . . . . . . . . . . . . 20 ({𝑧} ≠ ∅ → dom ((𝐴 ∖ ran 𝑔) × {𝑧}) = (𝐴 ∖ ran 𝑔))
3331, 32ax-mp 5 . . . . . . . . . . . . . . . . . . 19 dom ((𝐴 ∖ ran 𝑔) × {𝑧}) = (𝐴 ∖ ran 𝑔)
3430, 33ineq12i 4218 . . . . . . . . . . . . . . . . . 18 (dom 𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔))
35 disjdif 4472 . . . . . . . . . . . . . . . . . 18 (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔)) = ∅
3634, 35eqtri 2765 . . . . . . . . . . . . . . . . 17 (dom 𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅
37 funun 6612 . . . . . . . . . . . . . . . . 17 (((Fun 𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})) ∧ (dom 𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅) → Fun (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
3828, 36, 37sylancl 586 . . . . . . . . . . . . . . . 16 (𝑔:𝐵1-1𝐴 → Fun (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
3938adantl 481 . . . . . . . . . . . . . . 15 ((𝑧𝐵𝑔:𝐵1-1𝐴) → Fun (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
40 dmun 5921 . . . . . . . . . . . . . . . . . 18 dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
4129uneq1i 4164 . . . . . . . . . . . . . . . . . 18 (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
4233uneq2i 4165 . . . . . . . . . . . . . . . . . 18 (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
4340, 41, 423eqtr2i 2771 . . . . . . . . . . . . . . . . 17 dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
44 f1f 6804 . . . . . . . . . . . . . . . . . . 19 (𝑔:𝐵1-1𝐴𝑔:𝐵𝐴)
4544frnd 6744 . . . . . . . . . . . . . . . . . 18 (𝑔:𝐵1-1𝐴 → ran 𝑔𝐴)
46 undif 4482 . . . . . . . . . . . . . . . . . 18 (ran 𝑔𝐴 ↔ (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
4745, 46sylib 218 . . . . . . . . . . . . . . . . 17 (𝑔:𝐵1-1𝐴 → (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
4843, 47eqtrid 2789 . . . . . . . . . . . . . . . 16 (𝑔:𝐵1-1𝐴 → dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
4948adantl 481 . . . . . . . . . . . . . . 15 ((𝑧𝐵𝑔:𝐵1-1𝐴) → dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
50 df-fn 6564 . . . . . . . . . . . . . . 15 ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ↔ (Fun (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∧ dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴))
5139, 49, 50sylanbrc 583 . . . . . . . . . . . . . 14 ((𝑧𝐵𝑔:𝐵1-1𝐴) → (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴)
52 rnun 6165 . . . . . . . . . . . . . . 15 ran (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧}))
53 dfdm4 5906 . . . . . . . . . . . . . . . . . 18 dom 𝑔 = ran 𝑔
54 f1dm 6808 . . . . . . . . . . . . . . . . . 18 (𝑔:𝐵1-1𝐴 → dom 𝑔 = 𝐵)
5553, 54eqtr3id 2791 . . . . . . . . . . . . . . . . 17 (𝑔:𝐵1-1𝐴 → ran 𝑔 = 𝐵)
5655uneq1d 4167 . . . . . . . . . . . . . . . 16 (𝑔:𝐵1-1𝐴 → (ran 𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})))
57 xpeq1 5699 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
58 0xp 5784 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ × {𝑧}) = ∅
5957, 58eqtrdi 2793 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
6059rneqd 5949 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ran ∅)
61 rn0 5936 . . . . . . . . . . . . . . . . . . . . 21 ran ∅ = ∅
6260, 61eqtrdi 2793 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
63 0ss 4400 . . . . . . . . . . . . . . . . . . . 20 ∅ ⊆ 𝐵
6462, 63eqsstrdi 4028 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
6564a1d 25 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∖ ran 𝑔) = ∅ → (𝑧𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
66 rnxp 6190 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∖ ran 𝑔) ≠ ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
6766adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
68 snssi 4808 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝐵 → {𝑧} ⊆ 𝐵)
6968adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧𝐵) → {𝑧} ⊆ 𝐵)
7067, 69eqsstrd 4018 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
7170ex 412 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∖ ran 𝑔) ≠ ∅ → (𝑧𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
7265, 71pm2.61ine 3025 . . . . . . . . . . . . . . . . 17 (𝑧𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
73 ssequn2 4189 . . . . . . . . . . . . . . . . 17 (ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵 ↔ (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
7472, 73sylib 218 . . . . . . . . . . . . . . . 16 (𝑧𝐵 → (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
7556, 74sylan9eqr 2799 . . . . . . . . . . . . . . 15 ((𝑧𝐵𝑔:𝐵1-1𝐴) → (ran 𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
7652, 75eqtrid 2789 . . . . . . . . . . . . . 14 ((𝑧𝐵𝑔:𝐵1-1𝐴) → ran (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
77 df-fo 6567 . . . . . . . . . . . . . 14 ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴onto𝐵 ↔ ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ∧ ran (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵))
7851, 76, 77sylanbrc 583 . . . . . . . . . . . . 13 ((𝑧𝐵𝑔:𝐵1-1𝐴) → (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴onto𝐵)
79 foeq1 6816 . . . . . . . . . . . . . 14 (𝑓 = (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝐴onto𝐵 ↔ (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴onto𝐵))
8079spcegv 3597 . . . . . . . . . . . . 13 ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ Fin → ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴onto𝐵 → ∃𝑓 𝑓:𝐴onto𝐵))
8121, 78, 80syl2im 40 . . . . . . . . . . . 12 ((𝑔 ∈ Fin ∧ 𝐴 ∈ Fin) → ((𝑧𝐵𝑔:𝐵1-1𝐴) → ∃𝑓 𝑓:𝐴onto𝐵))
8281expcomd 416 . . . . . . . . . . 11 ((𝑔 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑔:𝐵1-1𝐴 → (𝑧𝐵 → ∃𝑓 𝑓:𝐴onto𝐵)))
8382com12 32 . . . . . . . . . 10 (𝑔:𝐵1-1𝐴 → ((𝑔 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧𝐵 → ∃𝑓 𝑓:𝐴onto𝐵)))
8414, 83syland 603 . . . . . . . . 9 (𝑔:𝐵1-1𝐴 → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧𝐵 → ∃𝑓 𝑓:𝐴onto𝐵)))
8584exlimiv 1930 . . . . . . . 8 (∃𝑔 𝑔:𝐵1-1𝐴 → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧𝐵 → ∃𝑓 𝑓:𝐴onto𝐵)))
8610, 85syl 17 . . . . . . 7 (𝐵𝐴 → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧𝐵 → ∃𝑓 𝑓:𝐴onto𝐵)))
8786adantl 481 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐵𝐴) → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧𝐵 → ∃𝑓 𝑓:𝐴onto𝐵)))
888, 9, 87mp2and 699 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵𝐴) → (𝑧𝐵 → ∃𝑓 𝑓:𝐴onto𝐵))
8988exlimdv 1933 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵𝐴) → (∃𝑧 𝑧𝐵 → ∃𝑓 𝑓:𝐴onto𝐵))
907, 89syl5 34 . . 3 ((𝐴 ∈ Fin ∧ 𝐵𝐴) → (∅ ≺ 𝐵 → ∃𝑓 𝑓:𝐴onto𝐵))
91903impia 1118 . 2 ((𝐴 ∈ Fin ∧ 𝐵𝐴 ∧ ∅ ≺ 𝐵) → ∃𝑓 𝑓:𝐴onto𝐵)
92913com23 1127 1 ((𝐴 ∈ Fin ∧ ∅ ≺ 𝐵𝐵𝐴) → ∃𝑓 𝑓:𝐴onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626   class class class wbr 5143   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  Fun wfun 6555   Fn wfn 6556  wf 6557  1-1wf1 6558  ontowfo 6559  cdom 8983  csdm 8984  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989
This theorem is referenced by:  fodomfib  9369
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