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Theorem ofoaid2 43348
Description: Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
Assertion
Ref Expression
ofoaid2 (((𝐴𝑉𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵m 𝐴)) → ((𝐴 × {∅}) ∘f +o 𝐹) = 𝐹)

Proof of Theorem ofoaid2
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 simpll 767 . 2 (((𝐴𝑉𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵m 𝐴)) → 𝐴𝑉)
2 onss 7803 . . . . . . 7 (𝐵 ∈ On → 𝐵 ⊆ On)
3 sstr 4003 . . . . . . . 8 ((ran 𝐹𝐵𝐵 ⊆ On) → ran 𝐹 ⊆ On)
43expcom 413 . . . . . . 7 (𝐵 ⊆ On → (ran 𝐹𝐵 → ran 𝐹 ⊆ On))
52, 4syl 17 . . . . . 6 (𝐵 ∈ On → (ran 𝐹𝐵 → ran 𝐹 ⊆ On))
65anim2d 612 . . . . 5 (𝐵 ∈ On → ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ On)))
7 df-f 6566 . . . . 5 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
8 df-f 6566 . . . . 5 (𝐹:𝐴⟶On ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ On))
96, 7, 83imtr4g 296 . . . 4 (𝐵 ∈ On → (𝐹:𝐴𝐵𝐹:𝐴⟶On))
10 elmapi 8887 . . . 4 (𝐹 ∈ (𝐵m 𝐴) → 𝐹:𝐴𝐵)
119, 10impel 505 . . 3 ((𝐵 ∈ On ∧ 𝐹 ∈ (𝐵m 𝐴)) → 𝐹:𝐴⟶On)
1211adantll 714 . 2 (((𝐴𝑉𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵m 𝐴)) → 𝐹:𝐴⟶On)
13 peano1 7910 . . 3 ∅ ∈ ω
14 fnconstg 6796 . . 3 (∅ ∈ ω → (𝐴 × {∅}) Fn 𝐴)
1513, 14mp1i 13 . 2 (((𝐴𝑉𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵m 𝐴)) → (𝐴 × {∅}) Fn 𝐴)
16 simp3 1137 . . . 4 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐴 × {∅}) Fn 𝐴)
17 simp2 1136 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐹:𝐴⟶On)
1817ffnd 6737 . . . 4 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐹 Fn 𝐴)
19 simp1 1135 . . . 4 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐴𝑉)
20 inidm 4234 . . . 4 (𝐴𝐴) = 𝐴
2116, 18, 19, 19, 20offn 7709 . . 3 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → ((𝐴 × {∅}) ∘f +o 𝐹) Fn 𝐴)
2216, 18jca 511 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → ((𝐴 × {∅}) Fn 𝐴𝐹 Fn 𝐴))
2322adantr 480 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → ((𝐴 × {∅}) Fn 𝐴𝐹 Fn 𝐴))
2419adantr 480 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → 𝐴𝑉)
25 simpr 484 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → 𝑎𝐴)
26 fnfvof 7713 . . . . 5 ((((𝐴 × {∅}) Fn 𝐴𝐹 Fn 𝐴) ∧ (𝐴𝑉𝑎𝐴)) → (((𝐴 × {∅}) ∘f +o 𝐹)‘𝑎) = (((𝐴 × {∅})‘𝑎) +o (𝐹𝑎)))
2723, 24, 25, 26syl12anc 837 . . . 4 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → (((𝐴 × {∅}) ∘f +o 𝐹)‘𝑎) = (((𝐴 × {∅})‘𝑎) +o (𝐹𝑎)))
28 fvconst2g 7221 . . . . . 6 ((∅ ∈ ω ∧ 𝑎𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
2913, 25, 28sylancr 587 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
3029oveq1d 7445 . . . 4 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → (((𝐴 × {∅})‘𝑎) +o (𝐹𝑎)) = (∅ +o (𝐹𝑎)))
3117ffvelcdmda 7103 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → (𝐹𝑎) ∈ On)
32 oa0r 8574 . . . . 5 ((𝐹𝑎) ∈ On → (∅ +o (𝐹𝑎)) = (𝐹𝑎))
3331, 32syl 17 . . . 4 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → (∅ +o (𝐹𝑎)) = (𝐹𝑎))
3427, 30, 333eqtrd 2778 . . 3 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → (((𝐴 × {∅}) ∘f +o 𝐹)‘𝑎) = (𝐹𝑎))
3521, 18, 34eqfnfvd 7053 . 2 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → ((𝐴 × {∅}) ∘f +o 𝐹) = 𝐹)
361, 12, 15, 35syl3anc 1370 1 (((𝐴𝑉𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵m 𝐴)) → ((𝐴 × {∅}) ∘f +o 𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wss 3962  c0 4338  {csn 4630   × cxp 5686  ran crn 5689  Oncon0 6385   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  f cof 7694  ωcom 7886   +o coa 8501  m cmap 8864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-oadd 8508  df-map 8866
This theorem is referenced by: (None)
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