Theorem fodju0 7222

Description: Lemma for fodjuomni 7224 and fodjumkv 7235. A condition which shows that 𝐴 is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)

Hypotheses

Ref	Expression
fodjuf.fo	⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
fodjuf.p	⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))
fodju0.1	⊢ (𝜑 → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)

Assertion

Ref	Expression
fodju0	⊢ (𝜑 → 𝐴 = ∅)

Distinct variable groups:   𝜑,𝑦,𝑧   𝑦,𝑂,𝑧   𝑧,𝐴   𝑧,𝐵   𝑧,𝐹   𝑦,𝐴   𝑦,𝐹   𝑤,𝑂   𝑤,𝑃

Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑤)   𝐵(𝑦,𝑤)   𝑃(𝑦,𝑧)   𝐹(𝑤)

Proof of Theorem fodju0

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fodjuf.fo	. . . . 5 ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
2		djulcl 7126	. . . . 5 ⊢ (𝑢 ∈ 𝐴 → (inl‘𝑢) ∈ (𝐴 ⊔ 𝐵))
3		foelrn 5802	. . . . 5 ⊢ ((𝐹:𝑂–onto→(𝐴 ⊔ 𝐵) ∧ (inl‘𝑢) ∈ (𝐴 ⊔ 𝐵)) → ∃𝑣 ∈ 𝑂 (inl‘𝑢) = (𝐹‘𝑣))
4	1, 2, 3	syl2an 289	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → ∃𝑣 ∈ 𝑂 (inl‘𝑢) = (𝐹‘𝑣))
5		fodjuf.p	. . . . . 6 ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))
6		fveqeq2 5570	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑦) = (inl‘𝑧) ↔ (𝐹‘𝑣) = (inl‘𝑧)))
7	6	rexbidv 2498	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧)))
8	7	ifbid 3583	. . . . . 6 ⊢ (𝑦 = 𝑣 → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o))
9		simprl 529	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → 𝑣 ∈ 𝑂)
10		peano1 4631	. . . . . . . 8 ⊢ ∅ ∈ ω
11	10	a1i 9	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∅ ∈ ω)
12		1onn 6587	. . . . . . . 8 ⊢ 1o ∈ ω
13	12	a1i 9	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → 1o ∈ ω)
14	1	fodjuomnilemdc 7219	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
15	14	ad2ant2r 509	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
16	11, 13, 15	ifcldcd 3598	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o) ∈ ω)
17	5, 8, 9, 16	fvmptd3 5658	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (𝑃‘𝑣) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o))
18		fveqeq2 5570	. . . . . 6 ⊢ (𝑤 = 𝑣 → ((𝑃‘𝑤) = 1o ↔ (𝑃‘𝑣) = 1o))
19		fodju0.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)
20	19	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)
21	18, 20, 9	rspcdva 2873	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (𝑃‘𝑣) = 1o)
22		simplr 528	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → 𝑢 ∈ 𝐴)
23		simprr 531	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (inl‘𝑢) = (𝐹‘𝑣))
24	23	eqcomd 2202	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (𝐹‘𝑣) = (inl‘𝑢))
25		fveq2 5561	. . . . . . . 8 ⊢ (𝑧 = 𝑢 → (inl‘𝑧) = (inl‘𝑢))
26	25	rspceeqv 2886	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝐹‘𝑣) = (inl‘𝑢)) → ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
27	22, 24, 26	syl2anc 411	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
28	27	iftrued 3569	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o) = ∅)
29	17, 21, 28	3eqtr3rd 2238	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∅ = 1o)
30	4, 29	rexlimddv 2619	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → ∅ = 1o)
31		1n0 6499	. . . . 5 ⊢ 1o ≠ ∅
32	31	nesymi 2413	. . . 4 ⊢ ¬ ∅ = 1o
33	32	a1i 9	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → ¬ ∅ = 1o)
34	30, 33	pm2.65da 662	. 2 ⊢ (𝜑 → ¬ 𝑢 ∈ 𝐴)
35	34	eq0rdv 3496	1 ⊢ (𝜑 → 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 835   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  ∅c0 3451  ifcif 3562   ↦ cmpt 4095  ωcom 4627  –onto→wfo 5257  ‘cfv 5259  1_oc1o 6476   ⊔ cdju 7112  inlcinl 7120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-1st 6207  df-2nd 6208  df-1o 6483  df-dju 7113  df-inl 7122  df-inr 7123

This theorem is referenced by:  fodjuomnilemres  7223  fodjumkvlemres  7234